Density mean back relaxation: Analytical solutions, properties, and application in experiments

Master’s Thesis Density mean back relaxation Analytical solutions, properties, and application in experiments Mittlere Rückrelaxation von Dichten Analytische Lösungen, Eigenschaften und Anwendung in Experimenten prepared by Laila Henkes from Stuttgart at the Institute for Theoretical Physics Thesis period: 10th March 2025 until 10th September 2025 First referee: Prof. Dr. Matthias Krüger Second referee: Prof. Dr. Timo Betz Abstract The breakdown of time reversibility is one of the key concepts that distinguishes non-equilibrium from equilibrium systems. However, detecting it from experiments can be challenging. One method that allows to do so in systems with a confining potential is the recently introduced mean back relaxation (MBR). In this thesis, we demonstrate that the MBR for densities (dMBR) is an extension of the MBR that enables the detection of broken time-reversal symmetry, even in systems without a confining potential. For systems with a finite number of particles that are described by a microscopic density, we additionally find a lower bound on entropy production. We provide analytical expressions of the dMBR for Gaussian example systems of both microscopic and macroscopic densities and find good agreement with results from simulations. Evaluating experimental data, we demonstrate that the dMBR can detect activity in biological cells with a confidence of 2σ. In addition, the derived entropy bounds detect non-zero entropy production. Finally, we look at the MBR of intensities obtained from scattering (iMBR), extending the applicability of the MBR to detect broken time-reversal symmetry to scattering experiments. ii Contents 1. Introduction 1 2. Concepts and methods 3 2.1. Stochasticprocesses............................ 3 2.1.1. The Langevin equation . . . . . . . . . . . . . . . . . . . . . . 3 2.1.2. The Fokker-Planck equation . . . . . . . . . . . . . . . . . . . 5 2.1.3. Equilibrium and detailed balance condition . . . . . . . . . . . 5 2.2. Path integration and minimum path weight . . . . . . . . . . . . . . 7 2.3. Mean back relaxation for positions . . . . . . . . . . . . . . . . . . . 8 2.3.1. Definition and properties . . . . . . . . . . . . . . . . . . . . . 8 2.3.2. MBR of two harmonically coupled particles . . . . . . . . . . 10 2.4. Mean back relaxation for densities . . . . . . . . . . . . . . . . . . . . 12 2.4.1. Density and its Fourier transform . . . . . . . . . . . . . . . . 12 2.4.2. Definition and properties of dMBR . . . . . . . . . . . . . . . 12 2.5. Time evolution of densities and the Dean-Kawasaki equation . . . . . 13 3. Density MBR for microscopic densities 15 3.1. Properties................................. 15 3.1.1. Density MBR is real for Gaussian systems . . . . . . . . . . . 15 3.1.2. Entropybounds.......................... 16 3.2. Density MBR for two harmonically coupled particles . . . . . . . . . 18 3.3. Horse-and-cart model . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.1. Introduction of the model . . . . . . . . . . . . . . . . . . . . 29 3.3.2. DensityMBR........................... 30 3.3.3. Scaling of the dMBR with the system parameters . . . . . . . 31 3.3.4. Detecting broken DB at D=Dq................. 35 3.3.5. Entropybounds.......................... 37 3.4. Vesiclesincells .............................. 40 3.4.1. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 40 iii Contents 3.4.2. DensityMBR........................... 41 3.4.3. Entropybounds.......................... 45 4. Density MBR for macroscopic densities 48 4.1. Single-component densities . . . . . . . . . . . . . . . . . . . . . . . . 48 4.1.1. Analytical solution for Gaussian density fluctuations . . . . . 48 4.1.2. Free Brownian particles . . . . . . . . . . . . . . . . . . . . . 50 4.2. Multi-component mixtures . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2.1. Analytical solution for Gaussian density fluctuations . . . . . 51 4.2.2. Two-component mixture with non-reciprocal interaction . . . . 52 5. Detecting broken DB from scattering experiments 54 5.1. IntensityandMBR............................ 54 5.2. Intensity MBR for two harmonically coupled particles . . . . . . . . . 55 5.3. Intensity MBR for two active Brownian particles . . . . . . . . . . . . 56 6. Conclusion 58 6.1. Summary ................................. 58 6.2. Outlook .................................. 60 A. Two harmonically coupled particles 69 A.1. Probability distribution for two harmonically coupled particles . . . . 69 A.2. Density MBR for two harmonically coupled particles with weighted density................................... 71 B. Density MBR and ISF 76 B.1.Singlecomponent............................. 76 B.2.Multiplecomponents ........................... 78 B.2.1. Derivation of the dMBR . . . . . . . . . . . . . . . . . . . . . 78 B.2.2. ISF of a two-component mixture . . . . . . . . . . . . . . . . 78 iv 1. Introduction How can we measure the vitality of a system? The first step is to distinguish between systems at equilibrium, which are definitely non-living, and non-equilibrium systems, which are possibly living [1]. Whenever time-reversal symmetry is broken or entropy is produced, a system is out of equilibrium [2]. Therefore, from a theoretical point of view, the challenge is to find an observable related to these two concepts to quantify how far a system is away from equilibrium. For entropy production, many powerful theorems and relations have been derived [2–6]. However, entropy is generally not operationally accessible in experiments due to limited resolution [5]. On the other hand, breakage of time-reversal symmetry can be captured by multi-point correlation functions, i.e., correlation functions that depend on multiple points in time [7]. A well-established approach to probe broken time-reversal symmetry is to use the fluctuation dissipation theorem (FDT) [8–12]. The FDT relates correlation functions of the motion of particles in equilibrium to the response of the system to external perturbations [2, 13–21]. This relation can be used by measuring the response of the system to a perturbation, usually by applying a force to a particle [11, 22, 23]. The equilibrium particle motion predicted by the FDT can then be compared with the actual particle motion, allowing non-equilibrium contributions to be detected [24]. Since today’s advanced techniques allow particle tracking with high precision and time resolution, the non-invasive measurement of the motion of the particle is easy to perform [25]. However, the invasive measurement needed to determine the response to external perturbations is more difficult. The use of the FDT requires the application of very small and controlled forces, making such measurements challenging [24, 26]. It is therefore an important and ongoing task to find new methods that allow the detection of non-equilibrium from non-invasive measurements alone. In the past years, several methods have been suggested to detect non-equilibrium behaviour non-invasively from the observation of particle trajectories. One method proposed is the observation of currents [24, 27, 28]. There exist many theorems, 1 1. Introduction e.g., the thermodynamic uncertainty relations [4, 5, 29], yielding a lower bound on entropy production depending on observed currents and their fluctuations [30, 31], and thus providing an estimate of how far away from equilibrium a system is. Another method is to look at probability distributions of certain times that can be extracted from trajectories. For systems with discrete states, the probability distribution of waiting times, i.e., the time a system spends in the same state, has been shown to be a marker of broken time-reversal symmetry [32, 33]. For continuous systems, the time to reach the maximum of a trajectory in a time interval of fixed length has been considered [34, 35]. It has been shown that the probability distribution for this time is symmetric around its maximum in equilibrium systems and can be antisymmetric for systems out of equilibrium [34, 35]. However, in some cases, e.g., for Gaussian systems, this method is unable to detect broken timereversal symmetry [34]. One multi-point observable that has proved useful in detecting non-equilibrium is the recently introduced mean back relaxation (MBR) [7, 26]. The MBR can be extracted from the passive observation of particle trajectories. It has been shown that MBR is a marker for broken time-reversal symmetry in arbitrary confined systems [26], which has been confirmed in experiments on both living and artificial active systems [26]. However, it can not be used as a marker for non-equilibrium in unconfined systems [7]. In such systems, the MBR of the Fourier transform of the density has been proposed as a possible marker for non-equilibrium [7]. The properties of this so-called density MBR (dMBR) will be investigated in this thesis. The thesis is structured as follows. In Ch. 2, the fundamental physical concepts and mathematical methods are introduced. Additionally, the limitation of the MBR to systems with a confining potential is exemplified, and the dMBR is defined. Properties of the dMBR for microscopic densities are discussed in Ch. 3. Furthermore, it is shown that the dMBR can detect broken time-reversal symmetry in two example systems where the position MBR cannot do so, i.e., for two harmonically coupled particles and the random horse-and-cart model. Finally, the dMBR is applied to experimental data from cells, where it also shows to be a useful tool to detect broken time-reversal symmetry. In Ch. 4, the dMBR for macroscopic densities is discussed. A potential extension of the MBR to intensities is introduced in Ch. 5, which makes the MBR applicable to detect time-reversal symmetry breakage from scattering experiments. The thesis is rounded off with a summary and an outlook in Ch. 6. 2 2. Concepts and methods 2.1. Stochastic processes Many of the systems that we want to describe physically - from biological systems to gases, liquids, solids, and even stellar matter - are made up of many particles [36]. In such multi-particle systems, it is usually not possible to observe all the particles. However, it is often sufficient to solely observe parts of the system [36]. The influence of the unobserved remainder of the system is modelled by introducing a stochastic force. As a result, the description of the system becomes stochastic, leading to a stochastic equation of motion and a probability distribution for the observed particle’s positions. The former is called the Langevin equation and will be discussed in this section, as will be the Fokker-Planck equation, which describes the dynamics of the corresponding probability distribution. 2.1.1. The Langevin equation The Langevin equation is a suitable model to describe a system coupled to a heat bath. Originally, it was derived for a colloidal particle in a solute, where only the colloidal particle is observed but not the solvent particles [36, 37]. The underlying assumptions are that (i) the bath is large compared to the system and, uninfluenced by the system, always in equilibrium at constant temperature T, and (ii) the degrees of freedom in the bath evolve much faster than the degrees of freedom of the system we observe [36]. Consider Nparticles, each of mass m, in one dimension1. Each particle is coupled to a bath at temperature Ti, i = 1, ..., N. The positions xiof the particles can be written as an N-dimensional vector x x x= (x1, ..., xN)T. We allow for interaction via a possibly time-dependent potential V(x x x, t), leading to a conservative force on the particles. In addition, we allow for a non-conservative force f f f(x x x, t), e.g., due to external driving of the system, such that the force on the particles can be 1With this method, we can equivalently describe, e.g., N/3particles in three dimensions. 3 2. Concepts and methods written as an N-dimensional vector F F F(x x x, t) = −∇V(x x x, t) + f f f(x x x, t).(2.1) In addition to this potential and external force, the bath affects the particles. On one hand, the bath provides friction in the form of a matrix γ γ γ, which leads to a friction force −γ γ γ˙ x x x, where the dot denotes temporal derivative. On the other hand, the fast-moving degrees of freedom exert a force on the particles in the system, which is modelled by introducing a stochastic variable ξifor each bath, the so-called noise. Summing up all the force contributions and using Newton’s second law, we obtain a stochastic equation of motion. This is called the Langevin equation [36–39] m¨ x x x+γ γ γ˙ x x x=F F F(x x x, t) + γ γ γξ ξ ξ(t),(2.2a) ⟨ξ ξ ξ(t)⊗ξ ξ ξ(t′)⟩= 2D D Dδ(t−t′),(2.2b) ⟨ξ ξ ξ(t)⟩= 0,(2.2c) where we introduced the diffusion matrix D D D. The properties of the bath are reflected in the cumulants of the noise. In this thesis, we will always consider noise that has zero mean and is delta-correlated in time, i.e., it is white noise [38]. Furthermore, the noise in this thesis is always assumed to be Gaussian, which means that it is fully described by its mean and variance [38]. All higher cumulants disappear. In general, the diffusivity of the system is given by [36] D D D=kBT T Tγ γ γ−1,(2.3) where T T T= diag(T1, ..., TN)is a matrix containing the temperatures of the different baths on the diagonal and kBis the Boltzmann constant. For a diagonal friction matrix γij =γiδij, this reduces to the Einstein relation [36, 40] Dij =kBTi γi δij.(2.4) The diffusion matrix is positive semidefinite [37]. In principle, the entries of the diffusion matrix can depend on the positions x x x. Then, the noise is called multiplicative noise, while for constant D D D, the noise is called additive [37]. One important special case of Eq. (2.2) is the overdamped limit. If the friction term γ γ γ˙ x x xis much larger than the inertia term m¨ x x x, we can neglect the latter [36]. 4 2. Concepts and methods This approximation holds for small particles, as is often the case, e.g., in biological systems [41]. In this limit, the Langevin equation reduces to a first-order stochastic differential equation, the so-called overdamped Langevin equation [36, 37] ˙ x x x=γ γ γ−1F F F(x x x, t) + ξ ξ ξ(t).(2.5) The noise cumulants are the same as stated in Eqs. (2.2). 2.1.2. The Fokker-Planck equation The Langevin equation describes how the positions of particles in a stochastic system evolve over time. This raises the question of how the probability of finding the particles at positions x x xat time t, i.e., P(x x x, t), evolves over time. The time evolution of P(x x x, t)is governed by the Fokker-Planck equation, which can be used equivalently to the Langevin equation to describe the same stochastic system [2]. For the overdamped system described by Eq. (2.5), it reads [36–39] ∂ ∂tP(x x x, t) = ∇·γ γ γ−1F F F(x x x, t)P(x x x, t)+∇·(∇·(D D DP(x x x, t))) .(2.6) Here, the divergence of a matrix A A Ais a vector defined as (∇·A A A)i=Pj∂xjAij. The Fokker-Planck equation can be written as a continuity equation [36, 37, 39] ∂ ∂tP(x x x, t) = −∇·j j j(x x x, t),(2.7) where j j j(x x x, t)is the probability current and the total probability RdNxP(x x x, t)=1is conserved for all t[39]. The probability current can be read of from Eq. (2.6) and is [37] j j j(x x x, t) = γ γ γ−1F F F(x x x, t)P(x x x, t)−∇·D D DP(x x x, t).(2.8) A system is stationary if the probability remains unchanged over time, or equivalently if ∇·j j j= 0 [37]. 2.1.3. Equilibrium and detailed balance condition An equilibrium system is defined as a stationary system in which j j j(x x x, t)=0for all x x xand t, i.e., all components of the probability current vanish. The system is then 5 2. Concepts and methods the case in any other unconfined system. Therefore, if we want to be able to detect broken DB also in systems without confinement, it is worth thinking about a new observable similar to the MBR that satisfies Eqs. (2.22) and, thus, can be used to detect broken DB in unconfined systems. One possible observable we investigate is the density MBR. 2.4. Mean back relaxation for densities 2.4.1. Density and its Fourier transform We start with the microscopic density of Npoint particles in ndimensions [45] ρ(x x x, t) = 1 N N X j=1 δ(x x x−x x xj(t)),(2.31) and take the Fourier transform [45] ρq q q(t) = 1 N N X j=1 eiq q q·x x xj(t),(2.32) where q q qis a n-dimensional wavevector. In the limit N→ ∞, the density ρ(x x x, t) becomes a smooth function of x x x, i.e., the macroscopic densities. For typical particle interactions, observations show that the expectation value of the microscopic density as well as the macroscopic density ⟨ρ(x x x, t)⟩is finite, and we have a well-defined stationary distribution [45]. For example for a homogeneous density, i.e., ⟨ρ(x x x, t)⟩= ρ0, we find that ⟨ρq q q(t)⟩=ρ0δ(q q q)which vanishes for finite q q q. Therefore, for the density and its Fourier transform, the conditions posed in Eqs. (2.22) are satisfied in stationary states. 2.4.2. Definition and properties of dMBR Thus, we can now define the dMBR [7] Mρ(τ, t, l,q q q) = *−ρq q q(t)−ρq q q(0) ρq q q(0) −ρq q q(−τ)ϑl(|ρq q q(0) −ρq q q(−τ)|)+.(2.33) 12 2. Concepts and methods Since Eqs. (2.22) are satisfied, we know that lim t→∞ Mρ(τ, t, l,q q q)DB =1 2,(2.34a) lim t→∞ (Mρ(τ, t, l,q q q) + Mρ r(τ, t, l,q q q)) =1.(2.34b) The dMBR seems to have all the desirable properties we had for the MBR in confinement, even for unconfined systems. This suggests that the dMBR may be seen as an extension of MBR that allows us to detect broken DB in unconfined systems by observing particle trajectories. 2.5. Time evolution of densities and the Dean-Kawasaki equation For macroscopic systems where we are not interested in or do not know the microscopic details, it is useful to have equations of motion for the density instead of the positions of all particles. It can be shown that the density for an n-dimensional system of interacting Brownian particles follows the Dean-Kawasaki equation [46] ∂tρ(x x x, t) = ∇· ρ(x x x, t)∇δF δρ(x x x, t)+q2Dρ(x x x, t)ξ ξ ξ(x x x, t)!.(2.35) Here, Fis a coarse-grained free energy functional and ξ ξ ξ(x x x, t)is Gaussian white noise in space and time with correlation function ⟨ξ ξ ξ(x x x, t)⊗ξ ξ ξ(x x x′, t′)⟩=δ(x x x−x x x′)δ(t−t′)I. The tensor product of two vectors is a matrix with entries (v v v⊗w w w)ij =viwj, and I is the identity matrix. In principle, the diffusivity Dis also allowed to depend on ρ(x x x). We can directly see that the density obeys a continuity equation, implying conservation of particle number. For particles interacting via a pairwise interaction potential V(x x x−y y y), the underlying microscopic dynamics of the positions x x xiof N particles is given by [46] ˙ x x xi=−1 γ∇ N X j=1 V(x x xi−x x xj) + ξ ξ ξi,(2.36a) ⟨ξ ξ ξi(t)⊗ξ ξ ξj(t′)⟩= 2Dδijδ(t−t′)I,(2.36b) ⟨ξ ξ ξi(t)⟩= 0.(2.36c) 13 2. Concepts and methods The free energy functional for this system is [46] F[ρ(·)] = DZdnxρ(x x x) log(ρ(x x x)) + 1 2Zdnxdnyρ(x x x)V(x x x−y y y)ρ(y y y).(2.37) With Eq. (2.37), the Dean-Kawasaki equation becomes [46] ∂tρ(x x x, t) = D∆ρ(x x x, t) + 1 γ∇·ρ(x x x, t)Zdnyρ(y y y, t)∇V(x x x−y y y) + ∇·q2Dρ(x x x, t)ξ ξ ξ(x x x, t). (2.38) It is, in principle, difficult to obtain a closed expression for the time evolution of the Fourier-transformed density due to the non-linearity and the multiplicative noise. However, if the density fluctuations ϕ= (ρ− ⟨ρ⟩)/q⟨ρ⟩are small, as is the case, e.g., for large particle number [47], the fluctuations can be assumed to be Gaussian as a consequence of the central limit theorem. In this limit, we can linearise the Dean-Kawasaki equation for the density fluctuations [47] ∂tϕ(x x x, t) = D∆ϕ+⟨ρ⟩ γ∆(V∗ϕ)(x x x, t) + ∇·q2D⟨ρ⟩ξ ξ ξ(x x x, t).(2.39) Here, we introduced the notation (V∗ϕ)(x x x, t)for the convolution RdnyV (x x x− y y y)ϕ(y y y, t). In the absence of an external potential or density gradient, the average density is constant in space. We can therefore Fourier transform both sides of the equation and obtain a time evolution equation for ϕq q q(t)directly ∂tϕq q q(t) = −q2Dϕq q q(t)−q2⟨ρ⟩ γˆ V(q q q)ϕq q q(t) + iq q q·q2D⟨ρ⟩ˆ ξ(q q q, t),(2.40) with n-dimensional wavevector q q q, and q=|q q q|. The Fourier transform of the Gaussian white noise is again Gaussian white noise with correlation function Dˆ ξ(q q q, t)ˆ ξ(q q q′, t′)E= (2π)dδ(t−t′)δ(q q q+q q q′).(2.41) Since ϕq q qand ρq q qonly differ by a factor and a constant, they yield the same dMBR. 14 3. Density MBR for microscopic densities 3.1. Properties 3.1.1. Density MBR is real for Gaussian systems For systems where the distribution of the particle positions is symmetric in the interparticle distances, it can be shown that the dMBR is purely real. For doing so, we introduce the density back relaxation in analogy to the back relaxation in Ref. [48], B(t) = −ρq q q(t)−ρq q q(0) ρq q q(0) −ρq q q(−τ)ϑl(|ρq q q(0) −ρq q q(−τ)|).(3.1) The average of this observable B(t)yields the dMBR as it is defined in Eq. (2.33). We can rewrite B(t)for a system with Nparticles as B(t) = −PN j=1 eiq q q·x x xj(t)−eiq q q·x x xj(0) PN j=1 eiq q q·x x xj(0) −eiq q q·x x xj(−τ) =−PN j,l=1 eiq q q·(x x xj(t)−x x xl(0)) +eiq q q·(x x xj(0)−x x xl(−τ)) −eiq q q·(x x xj(t)−x x xl(−τ)) −eiq q q·(x x xj(0)−x x xl(0)) PN j,l=1 eiq q q·(x x xj(0)−x x xl(0)) +eiq q q·(x x xj(−τ)−x x xl(−τ)) −eiq q q·(x x xj(0)−x x xl(−τ)) −eiq q q·(x x xj(−τ)−x x xl(0)) =−PN j,l=1 eiq q q·(x x xj(t)−x x xl(0)) +eiq q q·(x x xj(0)−x x xl(−τ)) −eiq q q·(x x xj(t)−x x xl(−τ)) −cos [q q q·(x x xj(0) −x x xl(0))] PN j,l=1 cos [q q q·(x x xj(0) −x x xl(0))] + cos [q q q·(x x xj(−τ)−x x xl(−τ))] −2 cos [q q q·(x x xj(0) −x x xl(−τ))]. (3.2) In the first line, we multiplied the numerator and denominator by the complex conjugate of the denominator. In the second step, we renamed summation indices for half of the sums in order to identify cosine functions. The dMBR is the average of the above quantity. Splitting the remaining eiq q q...-terms in the numerator into cosine and sine functions, we find that for distributions that are symmetric under 15 3. Density MBR for microscopic densities changing the sign of all interparticle distances x x xi(t)−x x xj(t′), the odd sine functions give zero mean. Since these are exactly the imaginary parts, we are left with only the real part after averaging. In Gaussian systems, the symmetry condition for the probability distribution is satisfied, and therefore, the dMBR in these systems is purely real. 3.1.2. Entropy bounds As discussed in Sec. 2.1.3, one consequence of broken time-reversal symmetry, i.e., broken DB, is entropy production. Therefore, we split our observable B(t)into time-reversal symmetric part Bs(t)and antisymmetric part Ba(t) Bs(t) = −1 2 ρq q q(t)−ρq q q(0) ρq q q(0) −ρq q q(−τ)ϑl(|ρq q q(0) −ρq q q(−τ)|) −1 2 ρq q q(−τ)−ρq q q(t−τ) ρq q q(t−τ)−ρq q q(t)ϑl(|ρq q q(t−τ)−ρq q q(t)|),(3.3a) Ba(t) = −1 2 ρq q q(t)−ρq q q(0) ρq q q(0) −ρq q q(−τ)ϑl(|ρq q q(0) −ρq q q(−τ)|) +1 2 ρq q q(−τ)−ρq q q(t−τ) ρq q q(t−τ)−ρq q q(t)ϑl(|ρq q q(t−τ)−ρq q q(t)|).(3.3b) Systems that obey DB are symmetric under time-reversal. As a consequence, the mean of any time-antisymmetric observable is zero, i.e. ⟨B(t)⟩DB =⟨Bs(t)⟩. We also know that for systems obeying DB, the long-time limit of the dMBR is 1/2. This matches the finding that the long-time limit of the symmetric part has an average of 1/2. For systems with broken DB, we can have a non-zero expectation value of Ba(t), and thus deviations of the dMBR from 1/2in the long-time limit. It has been shown that real observables Othat are antisymmetric under timereversal and bounded such that |O| ≤ 1allow to estimate entropy production [6] ⟨s⟩ ≥ ⟨O⟩log 1 + ⟨O⟩ 1−⟨O⟩!≥0.(3.4) By the assumptions made to derive this inequality, ⟨s⟩is the part of the entropy arising from dissipation into the environment [2]. We will consider example systems with overdamped dynamics in stationary states, for which this is the total change in entropy [6]. Our time-reversal antisymmetric observable Ba(t)is complex. For the bound to 16 3. Density MBR for microscopic densities be applicable, we have to extract a real observable from the complex observable Ba(t). Since both real and imaginary part have an upper limit, there are multiple possibilities to derive bounds for the entropy production. First, consider the real part Re (Ba(t)). This is a time-antisymmetric observable. Furthermore, we have |Re (Ba(t)) | ≤ 2N/l, leading directly to a lower bound on the entropy ⟨s⟩ ≥ l 2N⟨Re (Ba(t))⟩log 1 + l 2N⟨Re (Ba(t))⟩ 1−l 2N⟨Re (Ba(t))⟩!.(3.5) The same derivation holds with the imaginary part instead of the real part. However, for Gaussian systems we have seen that the dMBR is purely real and therefore ⟨Im (Ba(t))⟩= 0, such that in Gaussian systems the bound obtained by the imaginary part of Ba(t)is trivial. By construction, ⟨s⟩is the entropy produced in the time interval [−τ, t], i.e., during the entire trajectory used to evaluate the dMBR. The bound we find here is directly dependent on the cutoff parameter l, which must usually be chosen small to obtain sufficient statistics and to observe broken DB. Particularly, we need l < 2N, because otherwise the cutoff function ϑlis not well-defined. Therefore, we expect this bound to be small in many cases. Large particle numbers directly reduce the magnitude of the entropy estimate as well. This already shows that we only have a lower bound on the entropy and cannot capture the total entropy of the process, since we usually expect the entropy to increase with the number of degrees of freedom, see Eq. (2.12). Another possibility to obtain a real observable is to look at the absolute value of B(t). The antisymmetric part |B(t)|aof |B(t)|is |B(t)|a=|Bs(t) + Ba(t)|−|Bs(t)−Ba(t)|,(3.6) ⇒ |B(t)|a= ρq q q(t)−ρq q q(0) ρq q q(0) −ρq q q(−τ)ϑl(|ρq q q(0) −ρq q q(−τ)|) − ρq q q(−τ)−ρq q q(t−τ) ρq q q(t−τ)−ρq q q(t)ϑl(|ρq q q(t−τ)−ρq q q(t)|) (3.7) ≤2N l.(3.8) Therefore, we obtain the entropy bound ⟨s⟩ ≥ l 2N⟨|B(t)|a⟩log 1 + l 2N⟨|B(t)|a⟩ 1−l 2N⟨|B(t)|a⟩!.(3.9) 17 3. Density MBR for microscopic densities This looks very similar to the bound found in Eq. (3.5). The only difference is the observable Re (Ba(t)) =|B(t)|a. It remains to test which entropy bound leads to stronger conditions for which systems. But the flaws, considering the dependence on land N, are the same as for the bound with the real part (3.5). The final possibility discussed here to obtain a real and time-antisymmetric observable from Ba(t)is taking the sign of the real part of Re (Ba(t)). This observable is still antisymmetric under time-reversal and |sgn (Re [Ba(t)]) | ≤ 1, leading to a third and last lower bound on the entropy production ⟨s⟩≥⟨sgn (Re [Ba(t)])⟩log 1 + ⟨sgn (Re [Ba(t)])⟩ 1−⟨sgn (Re [Ba(t)])⟩!.(3.10) This bound has the advantage that it is not directly dependent on the cutoff parameter lor the particle number Nand therefore seems to be a more promising candidate to obtain sharper bounds on the entropy. The total entropy produced ⟨s⟩has to monotonically increase in time according to the second law of thermodynamics. As we have seen in Sec. 2.1.3, the entropy production rate in a stationary state for time independent force and diffusivity is constant in time, and thus the total entropy produced in the time interval [−τ, t] is ⟨s⟩= (t+τ)˙ Stot. This diverges linearly in tfor t→ ∞. Our bounds, on the other hand, do not diverge for large times, since ⟨Ba⟩converges to the deviation of the long-time dMBR from 1/2, which typically is a finite value. For long times, we therefore expect the bounds to be weak compared to the actual entropy production. Therefore, we can not find a direct connection between entropy production and longtime limit dMBR deviation from 1/2, i.e., limt→∞⟨Ba(t)⟩. But for shorter times, the entropy bounds derived in this section might be useful candidates for estimating the entropy. 3.2. Density MBR for two harmonically coupled particles As a first example system, we return to the system of two harmonically coupled Brownian particles without external potential in one dimension. This system is described by Eqs. (2.25). As we have seen in Sec. 2.3.2, the position MBR fails to detect broken DB since the system lacks confinement. This section aims to see that 18 3. Density MBR for microscopic densities the dMBR overcomes this problem and can detect broken DB for this unconfined system. Consider the system to be in the stationary state. To calculate the dMBR, we define the earlier density deviation d=ρq(0)−ρq(−τ), and the later density deviation b=ρq(t)−ρq(0), where qnow is a one-dimensional wavevector. We can rewrite the definition in Eq. (2.33) as Mρ(τ, t, l, q) = −ZCdddbb dϑl(|d|)P(b, d) = −ZCdddbb dϑl(|d|)P(b|d)P(d) =−ZCdd⟨ρq(t)−ρq(0)⟩|d dϑl(|d|)P(d).(3.11) In the first step, we used that the joint probability of the density deviations can be written in terms of the conditioned probability distribution P(b|d)and the marginal distribution P(d). Since band dare complex random variables, we have to integrate both over the complex plane. Integrating over byields the conditioned mean ⟨ρq(t)−ρq(0)⟩|d=⟨δ(ρq(0) −ρq(−τ)−d)(ρq(t)−ρq(0))⟩ ⟨δ(ρq(0) −ρq(−τ)−d)⟩.(3.12) First, we will calculate this. Then, plugging in the result in Eq. (3.11), we can determine the dMBR. The expectation values in the numerator and the denominator of Eq. (3.12) can both be determined by path integration (for the definition of an expectation value in the path integral formalism, see Eq. (2.15)). To obtain the conditioned mean, we calculate these two path integrals explicitly. The calculation is easier for the transformed variables xr=x1−x2, xs=γ1x1+γ2x2 γ1+γ2. Since our observable only depends on the path x x x(·)=(xr(·), xs(·)) at three different time points, it is sufficient to know the transition probabilities P(x x xt, t|x x xt′, t′)and the stationary distribution Ps(x x x). All intermediate time points of the path can be integrated out easily, since we are in a Gaussian system. The derivation of the transition probabilities and the stationary state distribution can be found in Ap. A.1, and the probability distributions are given in Eqs. (A.4). Given the probabilities, we can evaluate the numerator and the denominator of Eq. (3.12) separately. Prefactors will be neglected as they cancel when taking the 19 3. Density MBR for microscopic densities fraction. In the numerator, we have to solve the integral Zd2xtd2x0d2x−τδ(ρq(0) −ρq(−τ)−d)P(x x xt, t|x x x0,0)P(x x x0,0|x x x−τ,−τ)P(x x x−τ)(3.13) ×eiqxs,t eiq γ2 γ1+γ2xr,t +e−iq γ1 γ1+γ2x2,t −eiqxs,0eiq γ2 γ1+γ2xr,0+e−iq γ1 γ1+γ2xr,0. Since we have Gaussian probability distributions, we can solve the integral over the coordinates at time tand obtain Zd2x0d2x−τδ(ρq(0) −ρq(−τ)−d)P(x x x0,0|x x x−τ,−τ)P(x x x−τ) ×eiqxs,0 e−q2 4bt  e−q2(γ2 γ1+γ2−dt 2bt)2bt 4atbt−d2 teiqe−2k γµtγ2 γ1+γ2xr,0+e−q2(−γ1 γ1+γ2−dt 2bt)2bt 4atbt−d2 te−iqe−2k γµtγ1 γ1+γ2xr,0   −eiq γ2 γ1+γ2xr,0−e−iq γ1 γ1+γ2xr,0!.(3.14) In the denominator, we obtain the same result without the second and the third line. Next, we want to solve the integral over x x x−τ. To do so, we use the Fourier representation of the delta distribution and the series expansion of eeiqx . We have to keep in mind that we have a complex delta distribution here, meaning that we effectively have two delta distributions, one for the real and one for the imaginary part, which we can rewrite as δ(eiqx1,0+eiqx2,0−eiqx1,−τ−eiqx2,−τ−d) =Zdkdk′eikRe(eiqx1,0+eiqx2,0−eiqx1,−τ−eiqx2,−τ−d)eik′Im(eiqx1,0+eiqx2,0−eiqx1,−τ−eiqx2,−τ−d) =Zdkdk′e−ikRe(d)−ik′Im(d) ∞ X n,m,l,p=0 n′,m′,l′,p′=0 ∞ X ˜n, ˜m,˜ l,˜p=0 ˜n′,˜m′,˜ l′,˜p′=0 (3.15) × ik 2!n+m+l+p+n′+m′+l′+p′(−1)l+p+l′+p′ n!m!l!p!n′!m′!l′!p′!eiqx1,0(n−n′)eiqx2,0(m−m′)eiqx1,−τ(l−l′)eiqx2,−τ(p−p′) × k′ 2!˜n+ ˜m+˜ l+˜p+˜n′+ ˜m′+˜ l′+˜p′(−1)˜ l+˜p+˜n′+ ˜m′ ˜n! ˜m!˜ l!˜p!˜n′! ˜m′!˜ l′!˜p′!eiqx1,0(˜n−˜n′)eiqx2,0( ˜m−˜m′)eiqx1,−τ(˜ l−˜ l′)eiqx2,−τ(˜p−˜p′) =Zdkdk′e−ikRe(d)−ik′Im(d) ∞ X n,m,l,p=0 n′,m′,l′,p′=0 ∞ X ˜n, ˜m,˜ l,˜p=0 ˜n′,˜m′,˜ l′,˜p′=0 eixs,0qseixr,0qreixs,−τ˜qseixr,−τ˜qr × ik 2!n+m+l+p+n′+m′+l′+p′ k′ 2!˜n+ ˜m+˜ l+˜p+˜n′+ ˜m′+˜ l′+˜p′(−1)l+p+l′+p′ n!m!l!p!n′!m′!l′!p′! (−1)˜ l+˜p+˜n′+ ˜m′ ˜n! ˜m!˜ l!˜p!˜n′! ˜m′!˜ l′!˜p′!. 20 3. Density MBR for microscopic densities In the last step, we have introduced the shorthand notation qs=q(n−n′+m−m′+ ˜n−˜n′+ ˜m−˜m′),(3.16a) qr=q γ2 γ1+γ2 (n−n′+ ˜n−˜n′)−γ1 γ1+γ2 (m−m′+ ˜m−˜m′)!,(3.16b) ˜qs=ql−l′+p−p′+˜ l−˜ l′+ ˜p−˜p′,(3.16c) ˜qr=q γ2 γ1+γ2 (l−l′+˜ l−˜ l′)−γ1 γ1+γ2 (p−p′+ ˜p−˜p′)!.(3.16d) Inserting the delta distribution in this form into the integral allows us to solve the integration over xs,−τ, xr,−τ. We obtain in the denominator Zdkdk′Zdxs,0dxr,0e−cx2 r,0e−ikRe(d)−ik′Im(d) ∞ X n,m,l,p=0 n′,m′,l′,p′=0 ∞ X ˜n, ˜m,˜ l,˜p=0 ˜n′,˜m′,˜ l′,˜p′=0 × ik 2!n+m+l+p+n′+m′+l′+p′ k′ 2!˜n+ ˜m+˜ l+˜p+˜n′+ ˜m′+˜ l′+˜p′(−1)l+p+l′+p′ n!m!l!p!n′!m′!l′!p′! (−1)˜ l+˜p+˜n′+ ˜m′ ˜n! ˜m!˜ l!˜p!˜n′! ˜m′!˜ l′!˜p′! ×eiqxs,0(qs+˜qs)eiqxr,0qr+d 2b(1−e−2k γµτ)˜qs+e−2k γµτ˜qre−˜q2 s 4bτe−bτ(˜qr−d 2b˜qs)2 4aτbτ−d2 τ.(3.17) Integration over the coordinates at time −τdoes not change the contributions depending on time tand 0, i.e., the second and third line in Eq. (3.14). Therefore, we obtain the same result in the numerator with the only difference that we additionally have the second and third line of Eq. (3.14) in the integrand. Finally, we can solve the integral over xr,0. In the numerator, this yields Zdkdk′Zdxs,0e−ikRe(d)−ik′Im(d) ∞ X n,m,l,p=0 n′,m′,l′,p′=0 ∞ X ˜n, ˜m,˜ l,˜p=0 ˜n′,˜m′,˜ l′,˜p′=0 eiqxs,0(qs+˜qs+1)e−˜q2 s 4bτe−bτ(˜qr−d 2b˜qs)2 4aτbτ−d2 τ × ik 2!n+m+l+p+n′+m′+l′+p′ k′ 2!˜n+ ˜m+˜ l+˜p+˜n′+ ˜m′+˜ l′+˜p′(−1)l+p+l′+p′ n!m!l!p!n′!m′!l′!p′! (−1)˜ l+˜p+˜n′+ ˜m′ ˜n! ˜m!˜ l!˜p!˜n′! ˜m′!˜ l′!˜p′! × e−q2 4bt(e−q2(γ2 γ1+γ2−dt 2bt)2bt 4atbt−d2 te−(qs+dτ 2bτ(1−e−2k γµτ)˜qs+e−2k γµτ˜qr+qγ2 γ1+γ2e−k γµt)2 4c(3.18) +e−q2(−γ1 γ1+γ2−dt 2bt)2bt 4atbt−d2 te−(qs+dτ 2bτ(1−e−2k γµτ)˜qs+e−2k γµτ˜qr−qγ1 γ1+γ2e−k γµt)2 4c) −e−(qs+dτ 2bτ(1−e−2k γµτ)˜qs+e−2k γµτ˜qr+γ2 γ1+γ2)2 4c−e−(qs+dτ 2bτ(1−e−2k γµτ)˜qs+e−2k γµτ˜qr−γ1 γ1+γ2)2 4c!. 21 3. Density MBR for microscopic densities 0 2 4 6 8 10 12 t/1 k 0.0 0.2 0.4 0.6 ( , t,l,q) normal reversed = 0.1 1 k = 0.32 1 k = 1.0 1 k = 3.16 1 k = 10.0 1 k 0 2 4 6 8 10 12 10 2 10 1 eq2kB(1T1+2T2)/( 1+2)2t ( ) 1 lim t( , t,l,q) Figure 3.4.: Time evolution of the dMBR for different τas labeled, and q= 0.5qk/(kBT1)out of equilibrium for T2= 5T1. The cutoff parameter l= 10−4 is chosen small and γ1=γ2. The lines are obtained from numerically solving the integrals in Eqs. (3.25) and (3.26), while the markers show results from simulations. The logarithmic plot in the inset shows the scaling with which the dMBR reaches its long-time limit. the long-time limit, dMBR from normal and time-reversed dynamics add up to 1 and in equilibrium, the dMBR converges to 1/2for long times, as predicted by Eqs. (2.34). Out of equilibrium, the long-time limit of the dMBR deviates from 1/2. The deviation from 1/2seems to grow for smaller q. A logarithmic plot of the deviation of the dMBR from its long-time limit shows that the predicted exponential time scaling for large times is correct (see insets in Figs. 3.3 and 3.4). For very large t, we observe deviations from this scaling in the simulations and the numerical solution of the integrals. This originates from the finite computational accuracy, as the deviation from the long-time limit becomes very small. Additionally, for the simulations, longer times require longer trajectories, such that the statistic is worse for larger times compared to shorter times. Finally, let’s look at the limitation of Eqs. (3.25) and (3.26) to small cutoff parameters. As shown in Fig. 3.5, for larger l, we see deviations between simulations and the small-l-approximation for T1=T2. For T1=T2, we still see agreement between simulation and the numerical solution of Eq. (3.25), because in equilibrium 28 3. Density MBR for microscopic densities the conditioned mean is linear in d(see Fig. 3.1). 0 2 4 6 8 10 12 t /1 k 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ( , t , l , q ) equilibrium nonequilibrium simulation, l = 0.5 simulation, reversed theory, l 0 theory,reversed Figure 3.5.: Time evolution of the dMBR for T2=T1and q= 0.5q3k/(kBT1) (blue) and T2= 5T1and q= 0.5qk/(kBT1)(red) for a large cutoff parameter l= 0.5,τ=γ1/k, and γ1=γ2. 3.3. Horse-and-cart model 3.3.1. Introduction of the model The random horse-and-cart model describes a particle in a harmonic trap where the trap diffuses freely, independent of the particle’s position. This is a non-equilibrium system since the particle experiences a force from the trap, but the trap’s movement is independent of the particle. The Langevin equations for this system in one dimension are ˙x=−k γ(x−q) + ξx(t),(3.27a) ˙q=ξq(t),(3.27b) ⟨ξi(t)ξj(t′)⟩= 2Diδijδ(t−t′),(3.27c) ⟨ξi(t)⟩= 0.(3.27d) 29 3. Density MBR for microscopic densities Here, xis the position of the particle and qis the position of the minimum of the trap. The trap diffuses with a fixed diffusion constant Dq, while the particle has diffusion constant D=kBT/γ. The horse-and-cart model is a limiting case of the two harmonically coupled particles. We obtain it when letting T2, γ2→ ∞ while keeping Dq=kBT2/γ2fixed in Eqs. (2.25). 3.3.2. Density MBR Taking the limit T2→ ∞, γ2→ ∞, T2/γ2→Dqin our analytical solution, we obtain for the functions in the exponent of the probabilities defined in Eqs. (A.5) at→k2t 2kB1−e−k γtkt (T+γDq)1 + e−k γt−2γ2Dq1−e−k γt,(3.28a) bt→ k1 + e−k γt(T+γDq) 4kBDqkt (T+γDq)1 + e−k γt−2γ2Dq1−e−k γt,(3.28b) c→k 2kB(T+γDq),(3.28c) dt→γk kBkt (T+γDq)1 + e−k γt−2γ2Dq1−e−k γt.(3.28d) Assuming we can switch the order of the limit T2, γ2→ ∞ and the integration, we obtain from Eq. (3.25) Mρ(τ, t, l, q) =qc(4aτbτ−d2 τ) 2π3/2Zdxdydze−bτx2e−(4aτbτ−d2 τ) 4bτy2e−cz2 ×eiqz + 1 −e−q2 4bt(e−q2(1−dt 2bt)2bt 4atbt−d2 teiqe−k γtz+e−q2(dt 2bt)2bt 4atbt−d2 t) eiqz + 1 −eiq(x−dτ 2bτe−k γτy+dτ(1−e−2k γτ) 2bτz)(eiq(y+e−k γτz)+ 1) . (3.29) For the time-reversed dynamics, we take the same limit in Eq. (3.26), and obtain Mρ r(τ, t, l, q) = qc(4aτbτ−d2 τ) 2π3/2Zdxdydze−bτx2e−(4aτbτ−d2 τ) 4bτy2e−cz2(3.30) ×eiqz + 1 −e−q2 4bt(e−q2(1−dt 2bte−k γt)2bt 4atbt−d2 teiq(dt 2bt(1−e−2k γt)+e−k γt)z+e−q2(dt 2bte−k γt)2bt 4atbt−d2 teiq dt 2bt(1−e−2k γt)z) eiqz + 1 −eiq(x−dτ 2bτy)(eiq(y+e−k γτz)+ 1) . 30 3. Density MBR for microscopic densities Like for the model of two harmonically coupled particles, we can see good agreement between Eqs. (3.29) and (3.30) and simulations, as shown in Fig. 3.6. 0 2 4 6 8 10 12 t/k 0.2 0.0 0.2 0.4 0.6 0.8 ( , t,l,q) normal reversed = 0.1k = 0.32k = 1.0k = 3.16k = 10.0k 0 1 2 3 4 5 6 10 4 10 3 10 2 10 1 100 eq2Dqt ( ) 1 lim t( , t,l,q) Figure 3.6.: Time evolution of the dMBR for different τas labeled, and q= 0.5qk/(kBT)for Dq= 5D. The cutoff parameter l= 10−4is chosen small. The lines are obtained from numerically solving the integrals in Eqs. (3.29) and (3.30), while the markers show results from simulations. The logarithmic plot in the inset shows the scaling with which the dMBR reaches its long-time limit. 3.3.3. Scaling of the dMBR with the system parameters For long times, the dMBR will reach its long-time limit exponentially ∼exp(−q2Dqt). This behaviour is also found in simulations (see inset in Fig. 3.6). Apart from this, it is difficult to see the scaling with the various parameters of the dMBR and its deviation from 1/2from Eq. (3.29), because the integral is not solvable analytically. Even if we look at limiting cases like certain parameters going to 0or ∞, where the integrand could be expanded in these parameters, we will not obtain useful results, because in general, the limit and the integration will not commute. However, we can solve the integral numerically and investigate the behaviour with the various parameters. There are two aspects of this investigation. On the one hand, we want to see deviation from 1/2whenever DB is broken. Therefore, we want to know how to 31 3. Density MBR for microscopic densities tune the parameters we can choose when evaluating the dMBR, i.e., τ, l, and q, to make this deviation as large as possible. On the other hand, we want to know how far away from equilibrium our system is and whether the dMBR is an adequate measure for this. For this, it is interesting to look at the connection between the entropy production and the deviation of the long-time limit of the dMBR from 1/2. Starting with the first point, we want to know for which values of l, q and τ, the deviation from 1/2becomes largest. The behaviour of the dMBR with the cutoff parameter lcan only be investigated from simulations, since we only have theoretical expressions in the limit l→0. We find that the limit l→0seems to exist and shows the largest deviation from 1/2for all q(see Fig. 3.7). We can further see that for small l, small qlead to the strongest deviation from 1/2of the long-time dMBR. For larger l, however, we find that the deviation decreases faster for small q, such that there is a finite qmaximising the deviation from 1/2for finite l. The larger l, the larger also the qmaximising the deviation from 1/2. Since we are interested in maximising the deviation from 1/2, we will, from now on, consider lsmall. This also has the consequence that we can use Eqs. (3.29) and (3.30). 10 410 310 210 1100 l 0.0 0.2 0.4 0.6 0.8 1.0 1.2 |lim t ( , t , l , q )1 2| qk B T / k = 0.1 qk B T / k = 0.13 qk B T / k = 0.17 qk B T / k = 0.22 qk B T / k = 0.28 qk B T / k = 0.36 qk B T / k = 0.46 qk B T / k = 0.6 qk B T / k = 0.77 qk B T / k = 1.0 Figure 3.7.: Long-time limit of the dMBR as a function of the cutoff parameter l for different qas labeled at τ=γ/k, Dq/D = 5. For l→0, we find a monotonic behaviour of the deviation from 1/2in qas well. The deviation seems to diverge as 1/q for small q, as shown in Fig. 3.8. However, for smaller q, the dMBR also takes more time to reach its long-time limit, and the results are fluctuating more strongly, such that more data is needed. Therefore, we 32 3. Density MBR for microscopic densities have a trade-off between qand finite statistics. But in general, choosing q(and l) as small as possible seems to maximise the deviation of the long-time dMBR from 1/2for all model parameters k, γ, T, Dq. 0.0 0.2 0.4 0.6 0.8 1.0 q/k T 3 2 1 0 1 2 3 lim t( , t,l,q)1 2 normal reversed 10 1100 10 2 10 1 100 101 1/q Figure 3.8.: Deviation from 1/2of the long-time limit of the dMBR as a function of qfor τ=γ/k, l = 10−4, Dq/D = 5. The theoretical solution obtained from Eqs. (3.29) and (3.30) is marked by the red lines, the results from the simulation by the markers. In the inset, a double-logarithmic plot of the deviation from 1/2is shown. The choice of optimal τis more complicated and depends on the parameters of the model. Rewriting the system in a dimensionless form, we find that the only relevant parameters of the model are k/γ and Dq/D. The deviation of the long-time limit of the dMBR from 1/2as a function of these two parameters is shown in Fig. 3.9 for various values of τ. For k/γ, shown in Fig. 3.9(a), we find that decreasing τalways increases the deviation from 1/2. But the optimal τdepends non-trivially on Dq/D (see Fig. 3.9(b)). At the value of Dq/D where the deviation from 1/2is maximal, small τstill seems to be the best choice. But for larger and smaller Dq/D, larger τ leads to a stronger deviation from 1/2. Interestingly, for Dq=D, no deviation from 1/2is found for any τ. Regarding the second point, the connection between long-time dMBR deviation from 1/2and entropy production, we first need to calculate the entropy production rate of the system. Since we know the details of the system, this can be done 33 3. Density MBR for microscopic densities 0 50 100 150 200 k in 1/s 0 2 4 6 lim t ( , t , l , q ) = 0.1 s = 0.3 s = 1 s = 3.2 s = 10 s = 32 s = 100 s = 316 s (a) 10 1100101 Dq / D 0.2 0.4 0.6 0.8 lim t ( , t , l , q ) = 0.1 s = 0.3 s = 1 s = 3.2 s = 10 s = 32 s = 100 s = 316 s (b) Figure 3.9.: Long-time limit of the dMBR obtained from Eq. (3.29) as a function of the two relevant parameters of the system, i.e., k/γ with Dq= 5Dfixed in (a), and Dq/D with k/γ = 1/sfixed in (b). The value of τis as labeled, and q= 0.5/m. analytically. For doing so, we first calculate the probability current given by Eq. (2.8) j j j(x, q) = −kDq γ1(D+Dq)(x−q)Ps(x, q)(ˆ e e ex+ˆ e e eq).(3.31) The stationary state probability is P(x, q) = exp(−c(x−q)2)/Z, with cfor the horse-and-cart model given by Eq. (3.28c). In the stationary state of the system, 34 3. Density MBR for microscopic densities the total entropy production rate is given by Eq. (2.12). Plugging in the probability current (3.31), we find ˙ Stot =k γ Dq D=kDq T1 .(3.32) The entropy production rate is thus linear in the two relevant parameters of the system. It vanishes in two cases: (i) if the trap is not diffusing, i.e., Dq= 0, corresponding to the equilibrium case of a particle diffusing in a harmonic trap, and (ii) if the two particles decouple, i.e., k= 0. In the latter case, we have two freely diffusing particles (possibly at different temperatures), which is also an equilibrium system. Having in mind that the entropy production is linear in k/γ and Dq/D, we can see that the magnitude of the deviation of the long-term dMBR is not directly related to entropy production. For example, for Dq=D, the dMBR will always converge to 1/2(see Fig. 3.9(b)), even though DB is broken and entropy is produced. Also, for larger Dq/D, the entropy production increases while the deviation of the long-time dMBR from 1/2decreases. The same holds for large k/γ (see Fig. 3.9(a)). So, the dMBR can be used as a marker for broken DB because deviation of the long-term limit tells us that DB is broken. But the magnitude of this deviation seems not to be directly connected to entropy production. This means that we have to be careful when trying to quantify the activity of systems by using the dMBR. 3.3.4. Detecting broken DB at D=Dq For D=Dq, the dMBR converges towards 1/2for all values of τ,l, and q, even though DB is broken and entropy produced. This poses the question of whether there is a different observable that is able to detect breakage of DB in this situation. One candidate here is the position MBR (2.20). Even though for this unconfined system, Eqs. (2.23) do not apply, it has been shown that for the particular case of the horse-and-cart model, the MBR converges towards 1/2(1 −Dq/D)for k= 0 [26]. Therefore, we find that the position MBR deviates from 1/2if and only if DB is broken (Dq= 0). However, this is not a completely satisfying solution. One downside is that we are not able to see any difference between normal and timereversed dynamics. Furthermore, the same problem arises in the more general model of two harmonically coupled particles described by Eqs. (2.25). Here, we can also see no deviation from 1/2in the long-time limit of the dMBR if T1/γ1=T2/γ2, even 35 3. Density MBR for microscopic densities if T1=T2, i.e., if DB is broken. In this case, the position MBR is not able to detect broken DB. It might deviate from 1/2, but it can also deviate from 1/2if DB is obeyed and therefore yields no information about DB, as shown in Sec. 2.3.2. Another possible solution is to use different weights in the density ρ(x, t) = Pjwjδ(x−xj(t)). The weighted density could, e.g., describe the mass density of particles with different masses mi=wi. The weights are not affected by taking the Fourier transform, i.e., ρq(t) = Pjwjeiqxj(t). For the two-particle systems we looked at so far, this leads to different prefactors w1, w2in front of the exponentials eiqx1and eiqx2. Since we only have two particles, we can divide the numerator and denominator by one of the weights when calculating the dMBR. This leaves us with only one effective weight w=w2/w1. The dMBR for a weighted density can be calculated for small cutoff parameters lthe same way as it has been done in Sec. 3.2. The details of the derivation can be found in Ap. A.2. The weighted dMBR for the horse-and-cart model becomes Mρ(τ, t, l, q, w) =qc(4aτbτ−d2 τ) 2π3/2Zdxdydze−bτx2e−(4aτbτ−d2 τ) 4bτy2e−cz2(3.33) ×eiqz +w−e−q2 4bt(e−q2(1−dt 2bt)2bt 4atbt−d2 teiqe−k γtz+we−q2(dt 2bt)2bt 4atbt−d2 t) eiqz +w−eiq(x−dτ 2bτe−k γτy+dτ(1−e−2k γτ) 2bτz)(eiq(y+e−k γτz)+w) , which additionally to the parameters τ,l, and qalso depends on the relative weight w. For the time-reversed dynamics, we obtain Mρ r(τ, t, l, q, w) = qc(4aτbτ−d2 τ) 2π3/2Zdxdydze−bτx2e−(4aτbτ−d2 τ) 4bτy2e−cz2(3.34) ×eiqz +w−e−q2 4bt(e−q2(1−dt 2bte−k γt)2bt 4atbt−d2 teiq(dt 2bt(1−e−2k γt)+e−k γt)z+we−q2(dt 2bte−k γt)2bt 4atbt−d2 teiq dt 2bt(1−e−2k γt)z) eiqz +w−eiq(x−dτ 2bτy)(eiq(y+e−k γτz)+w) . The numerical solutions of Eqs. (3.33) and (3.34) are in good agreement with simulations (see Fig. 3.10). Additionally, we can see that the long-time limit of the dMBR deviates from 1/2for Dq=D. The same method also works for the two harmonically coupled particles at D1=D2. 36 3. Density MBR for microscopic densities 0246810 t / k 0.0 0.1 0.2 0.3 0.4 0.5 ( , t , l , q , w ) equilibrium, Dq = 0 nonequilibrium, Dq = D normal reversed Figure 3.10.: Time evolution of the dMBR of a weighted density for w= 2, τ = γ/k, l = 10−4, q =qk/(kBT)in equilibrium (Dq= 0) and out of equilibrium for Dq=D. 3.3.5. Entropy bounds Next, we discuss the entropy bounds derived in Sec. 3.1.2. For the horse-and-cart model, we know how much entropy is produced. We can therefore compare the lower bounds we obtain from Eqs. (3.5), (3.9) and (3.10) to the actual entropy production. Since we are in a steady state, the entropy production rate in Eq. (3.32) is constant in time such that the total entropy produced in the time interval [−τ, t]is given by ⟨s⟩=k γ Dq D(t+τ).(3.35) The comparison of Eq. (3.35) to the entropy bounds is shown in Fig. 3.11. Since we have no analytic expressions for the entropy bounds of this model system, these are results obtained from simulations. First of all, we notice that in equilibrium, the results of the entropy bounds are 4−5sizes of order smaller than out of equilibrium. In equilibrium, there is zero entropy production, such that the lower bounds should be zero. However, due to finite statistics and computational accuracy, we still get finite bounds numerically, but they are so small that we can interpret them as zero entropy production. Secondly, when comparing the three bounds, we find that the signum bound (3.10) is the strongest, followed by the bound with real part (3.5). The bound with 37 3. Density MBR for microscopic densities With this in mind, we look at the dMBR for short τwhere the long-time limit of the dMBR is already reached after 10 s. Unfortunately, then the deviation from 1/2is often rather small. However, for specific combinations of qand τ, we can see deviations of the long-time limit from 1/2that are larger than 1σ. This can either be the case if the real part does not converge to 1/2, or if the imaginary part does not converge to 0. In Fig. 3.15, we can see that both these cases are realised in the active cells. For comparison, the dMBR from the passivated cells is plotted as well, for which we can observe the expected convergence towards 1/2. For the active cells, we find that once the long-time limit is reached, it might deviate significantly from 1/2, and dMBR from normal and time-reversed dynamics add up to 1, which includes that the imaginary parts add up to 0. Lastly, we will compare the dMBR of three vesicles to the dMBR of a single vesicle. Calculating the dMBR of a single vesicle, we see again clear differences between normal and time-reversed dynamics in active cells (see Fig. 3.16). For small q, we also see differences between normal and time-reversed dynamics of the passivated cells. This might hint at a rest of activity in the passivated cells, which are not completely at equilibrium. But the differences are much smaller than for the active cells. 0 2 4 6 8 10 t in s 0.0 0.1 0.2 0.3 0.4 0.5 Re( ( , t,l,q)) q= 0.002 1 nm q= 0.011 1 nm q= 0.021 1 nm normal reversed 0 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 q= 0.002 1 nm q= 0.011 1 nm q= 0.021 1 nm Figure 3.16.: Time evolution of the real and the imaginary part of the dMBR of one vesicle for active (red) and passivated (blue) HeLa cells for different qas labeled, τ= 67.2 ms, and l= 0.0026. The coloured bands mark the 1σ-interval. In Fig. 3.17, the dMBR of one and three vesicles is compared. We find that there are no significant differences for qmuch larger or much smaller than the inverse of 44 3. Density MBR for microscopic densities 0 2 4 6 8 10 t in s 0.0 0.1 0.2 0.3 0.4 0.5 Re( ( , t,l,q)) q= 0.0001 1 nm q= 0.0021 1 nm q= 0.0115 1 nm 3 vesicles 1 vesicle Figure 3.17.: Time evolution of the real part of the dMBR for one and three vesicles for different qas labeled, τ= 50.8 ms, and l= 0.0026. The coloured bands mark the 1σ-interval. the typical distances between the vesicles. However, for q= 0.0021/nm, i.e., if 2π/q is of the size of the distances between the vesicles, the dMBR for one and three vesicles deviate strongly. Here, they only have overlap in the 1.97σ-interval, hinting at the different vesicles interacting and influencing one another even though they are separated by several µm (see Fig. 3.13). 3.4.3. Entropy bounds The entropy bounds derived in Sec. 3.1.2 can also be calculated from the trajectories of the vesicles. Since we now have a non-vanishing imaginary part, we obtain four different bounds: Eq. (3.5) with real part and the same equation with Im (Ba(t)) instead of Re (Ba(t)), Eq. (3.9), and Eq. (3.10). All four bounds will be discussed in the following. First of all, we find that all bounds are much lower for the passivated cells than for the active ones, as shown in Figs. 3.18, 3.19, 3.20, and 3.21. This is expected because in the passive cells, much less entropy is produced. More importantly, we do detect entropy production in the active cells. Since entropy production implies broken DB, we can conclude that DB is broken in the active cells from these observations. The largest bound can be obtained from the imaginary part and is shown in Fig. 3.18. As for the horse-and-cart model in Sec. 3.3.5, increasing the cutoff l 45 3. Density MBR for microscopic densities increases the bounds with real (and imaginary) part and with absolute value. However, for too large l, we cut away most of our data, especially for small qand τ. Consequently, the expectation values become strongly fluctuating and unreliable. 0246810 t + in s 0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030 entropy bound 0246810 t + in s 20 40 60 in ms Figure 3.18.: Entropy bound with imaginary part Eq. (3.5) (with Im[Ba(t)] instead of Re[Ba(t)]) for different τindicated by the colour and l= 0.275, q = 1.46/µmin active (left) and passivated (right) HeLa cells. The bound with real part Eq. (3.5) fluctuates stronger than the bound with imaginary part (see Fig. 3.19). Because of the stronger fluctuations, we can not choose las large as for the imaginary part and therefore obtain a much smaller bound. Nevertheless, we detect more entropy in the active than in the passivated cells. But to obtain more accurate values for the entropy bound, more data is needed. 0246810 t + in s 0.00 0.25 0.50 0.75 1.00 1.25 1.50 entropy bound 1e 6 0246810 t + in s 20 40 60 in ms Figure 3.19.: Entropy bound with real part Eq. (3.5) for different τindicated by the colour and l= 0.275, q = 14.6/µmin active (left) and passivated (right) HeLa cells. The bound with absolute value Eq. (3.9) fluctuates too strongly for large cutoff parameter lto make any statement. For small l, however, it fluctuates significantly less than the other three bounds (see Fig. 3.20). But in this regime, the entropy bound is very small. 46 3. Density MBR for microscopic densities 0246810 t + in s 0.0 0.2 0.4 0.6 0.8 1.0 entropy bound 1e 7 0246810 t + in s 20 40 60 in ms Figure 3.20.: Entropy bound with absolute value Eq. (3.9) for different τindicated by the colour and l= 0.01, q = 11.46/µmin active (left) and passivated (right) HeLa cells. The bound with signum Eq. (3.10) shown in Fig. 3.21 fluctuates very strongly. Smaller land qlead to higher values of the bound. But it is difficult to tell how large the entropy bound actually is because of the large fluctuations. The increase might as well just be caused by increasing fluctuations. To discuss this bound, more statistic is needed. 0246810 t + in s 0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 entropy bound 0246810 t + in s 20 40 60 in ms Figure 3.21.: Entropy bound with signum Eq. (3.10) for different τindicated by the colour and l= 10−4, q = 2.08/µmin active (left) and passivated (right) HeLa cells. Generally, we find that for all bounds, the estimated entropy is rather small. This is partially because we only observe three vesicles, i.e., a small part of the cell, and therefore only have access to a small part of the entropy produced in the cell. But as we found for the horse-and-cart model in Sec. 3.3.5, the bounds don’t catch all of the entropy produced in the observed part of the system. Therefore, it is expected that, also here, we only observe a part of the entropy produced by the movement of the three vesicles. Nonetheless, we are clearly able to detect that entropy is produced and therefore DB is broken. 47 4. Density MBR for macroscopic densities 4.1. Single-component densities 4.1.1. Analytical solution for Gaussian density fluctuations For this chapter, we consider systems described by macroscopic densities. For homogeneous systems with Gaussian distributed density fluctuations, we are able to calculate the dMBR analytically. There seems to be a variety of processes where the density fluctuations are assumed to be Gaussian, e.g., stochastic systems in equilibrium that macroscopically follow hydrodynamic equations [56, 57], or a system of many interacting Brownian particles [58]. We consider an n-dimensional systems with density ρ(x x x, t). The probability of a configuration of densities is then given by P[ρ(·,·)|ρ(x x x, 0)] ∼exp Zdnxdnydtdt′(ρ(x x x, t′)−ρ(x x x, 0))a(x x x,y y y, t −t′)(ρ(y y y, t)−ρ(y y y, 0)). (4.1) Note that we do not assume locality or Markovianity. Assuming additionally that the system is homogeneous, i.e., the correlation kernel fulfils a(x x x,y y y, t) = a(x x x−y y y, t), we can calculate the probability distribution for the Fourier-transformed density fluctuations P[ˆρ(·,·)|ˆρ(q q q, 0)] ∼exp −1 4π2Zdnxdnydtdt′dnqdnq′ˆ ϕ(q q q, t)e−iq q q·x x xa(x x x−y y y, t −t′)ˆ ϕ(q q q′, t′)e−iq q q′·y y y, (4.2) where ϕ(x x x, t) = ρ(x x x, t)−ρ(x x x, 0) and ˆ fis the Fourier transform of f. Substituting y y y′=x x x−y y y, we obtain P[ˆρ(·,·)|ˆρ(q q q, 0)] ∼exp −1 4π2Zdnxdny′dtdt′dnqdnq′ˆ ϕ(q q q, t)ˆ ϕ(q q q′, t)a(y y y′, t −t′)eiq q q·y y y′e−i(q q q+q q q′)·x x x = exp −1 2πZdtdt′dnqˆa(q q q, t −t′)ˆ ϕ(q q q, t)ˆ ϕ(−q q q, t′).(4.3) 48 4. Density MBR for macroscopic densities In the second step, we used that the integration over x x xyields a delta distribution δ(q q q+q q q′). Because ρ(x x x, t)and thus also ϕ(x x x, t)are real, we have ˆ ϕ(−q q q, t) = ˆ ϕ(q q q, t)∗. As a consequence, the real part of ˆ ϕ(q q q, t)is symmetric in q q qwhile the imaginary part is antisymmetric in q q q. Moreover, a(x x x, t)is a correlation kernel and therefore symmetric in x x x. Consequently, its Fourier transform is real and symmetric in q q q. Using these symmetries, we obtain P[ˆρ(·,·)|ˆρ(q q q, 0)] ∼exp"−1 2πZdtdt′dnqˆa(q q q, t −t′) Re(ˆ ϕ(q q q, t))Re(ˆ ϕ(q q q, t′)) +Im(ˆ ϕ(q q q, t))Im(ˆ ϕ(q q q, t′))!#. (4.4) The single Fourier modes decouple. Splitting the integral over q q qinto small parts, we see that for every fixed q q q,ρq q q(t)−ρq q q(0) is Gaussian distributed with zero mean. Furthermore, the real and the imaginary part are uncorrelated, while the variances of the real and the imaginary part are the same. Since the real and the imaginary parts of every single mode are independent one-dimensional Gaussian processes, they are time-reversal symmetric [59]. And since all modes are decoupled, the total process is time-reversal symmetric as well. Therefore, every system with homogeneous density distribution and Gaussian density fluctuations obeys DB, and the dMBR will converge to 1/2. Adapting the derivation for the MBR-MSD formula (2.24) in Ref. [7] to complex variables (for the details, see Ap. B.1), and using the correlation properties arising from Gaussian density distributions, we can relate the dMBR to the intermediate scattering function (ISF) ˆ S(q q q, t) = ⟨ρq q q(t)ρ−q q q(0)⟩.(4.5) The dMBR for Gaussian density fluctuations in homogeneous systems then reads Mρ(τ, t, l,q q q) = 1 2 1−Re ˆ S(q q q, t +τ)−ˆ S(q q q, t) Re ˆ S(q q q, τ)−ˆ S(q q q, 0) .(4.6) According to the Onsager regression hypothesis, the correlation function ˆ S(q q q, t) = ⟨ρq q q(t)ρ−q q q(0)⟩obeys the same time evolution as the decay of an imposed initial condition [60, 61]. Consequently, the ISF converges in the limit t→ ∞. Thus, the dMBR of homogeneous systems with Gaussian density fluctuations will always converge to 1/2, as expected for equilibrium systems. 49 4. Density MBR for macroscopic densities For hard-core lattice gases [62] as well as for free [63] and interacting Brownian particles [58], ˆ Sis known on scales much larger than the underlying microscopic dynamics. In the hydrodynamic limit, i.e., for times large compared to the mean collision time and length scales large compared to the mean free path of particles (accordingly small qcompared to the inverse length scale), we have [58, 62, 63] ˆ S(q q q, t)∝e−q q q·D D Dq q qt,(4.7) where D D Dis the bulk diffusivity matrix. This leads to a dMBR of Mρ(τ, t, l,q q q) = 1 21−e−q q q·D D Dq q qt,(4.8) independent of τand l. The bulk diffusivity is known for some lattice gas models, e.g., for the symmetric exclusion dynamics, it is I[62]. For interacting Brownian particles in a bath, it is D0⟨ρ⟩/(2χ)I, where χis the static compressibility and D0= kBT/γ the free diffusion coefficient [62]. For non-interacting Brownian particles, it is simply D0I[63]. 4.1.2. Free Brownian particles To validate Eq. (4.8), we look at the density of many non-interacting Brownian particles in one dimension. Assume that we have sufficiently many particles, such that density fluctuations are small compared to the mean density. Then, the time evolution of the Fourier-transformed density fluctuations is well approximated by the linearised Dean-Kawasaki equation (2.40) without potential ∂tϕq(t) = −q2Dϕq(t) + iqq2D⟨ρ⟩ˆ ξ(q, t).(4.9) Simulation of Eq. (4.9) shows indeed the expected exponential convergence of the dMBR towards 1/2according to Eq. (4.8). This is shown in Fig. 4.1 for different values of τand l. As expected, we also find that the dMBR is independent of these two parameters 50 4. Density MBR for macroscopic densities 0246810 q 2 Dt 0.0 0.1 0.2 0.3 0.4 0.5 ( , t , l , q ) simulation simulation, reversed = 1/( q 2 D ), l = 10 3 = 5/( q 2 D ), l = 10 2 = 3/( q 2 D ), l = 10 1 theory Figure 4.1.: Time evolution of the dMBR for the macroscopic density of free Brownian particles at equilibrium for different values of τ, l as labeled. 4.2. Multi-component mixtures 4.2.1. Analytical solution for Gaussian density fluctuations So far, we have seen that for homogeneous systems with Gaussian density fluctuations, we are able to calculate the dMBR for macroscopic densities analytically. However, since these are always equilibrium systems, the dMBR always converges to 1/2. This is because ˆ ϕ(q q q, t)are one-dimensional Gaussian random variables, such that the process always obeys DB. In multi-component mixtures, ˆ ϕ(q q q, t)is a multi-dimensional vector and DB might be broken, e.g., for two species at different temperatures [64, 65], or non-reciprocal interaction between different species [66, 67]. We still assume that density fluctuations are Gaussian, i.e., Eqs. (4.1) and (4.4) for the probability distributions are valid. The only difference is that now ρ(x x x, t)and consequently ρq q q(t)are vectors where each component ρ(i) q q q(t)describes one mixture component. The correlation kernels a(x x x, t)and ˆa(q q q, t)are matrices. Since we are not able to divide by a complex vector of density deviations, we have to define what the dMBR should be for a multi-dimensional density. Therefore, we introduce Mρ ij(τ, t, l,q q q) = *−ρ(i) q q q(t)−ρ(i) q q q(0) ρ(j) q q q(0) −ρ(j) q q q(−τ)ϑl(|ρ(j) q q q(0) −ρ(j) q q q(−τ)|)+,(4.10) where i, j denote the different components of the mixture. Similarly, we define the 51 4. Density MBR for macroscopic densities multi-component ISF [68] ˆ S(ij)(q q q, t) = Dρ(i) q q q(t)ρ(j) −q q q(0)E.(4.11) Again, only real parts are correlated to real parts and imaginary parts to imaginary parts in Eq. (4.4). But if the different species interact, i.e., if ˆa(q q q, t)has off-diagonal entries, the different components of ϕq q qare correlated, and we no longer have independent one-dimensional Gaussian processes. Adapting the derivation in Ap. B.1, as done in Ap. B.2.1, we find Mρ ij(τ, t, l,q q q) = 1 2 Re ˆ S(ij)(q q q, τ)−ˆ S(ij)(q q q, 0) + ˆ S(ij)(q q q, t)−ˆ S(ij)(q q q, t +τ) Re ˆ S(jj)(q q q, τ)−ˆ S(jj)(q q q, 0).(4.12) Since ˆ S(ij)(q q q, t)converges for t→ ∞, the diagonal entries Mρ ii converge to 1/2in the long-time limit. But in some cases, we might be able to detect broken DB from the off-diagonal entries. Looking at the time-reversal of ˆ S(ij)(t), we find that if DB holds, Re ⟨ρ(i) q q q(t)ρ(j) −q q q(0)⟩DB = Re ⟨ρ(i) q q q(0)ρ(j) −q q q(t)⟩= Re ⟨ρ(i) −q q q(0)ρ(j) q q q(t)⟩.(4.13) Therefore, if DB holds, Re( ˆ S(ij)(q q q, −t)) = Re( ˆ S(ij)(q q q, t)). If additionally ˆ S(ii)(q q q, t) = ˆ S(jj)(q q q, t), i.e., Mρ ii =Mρ jj, we can deduce broken DB if we find that Mρ ij(τ, t, l,q q q)= Mρ ji(τ, t, l,q q q)for one pair of i, j. One example system where this works will be discussed in the next section. 4.2.2. Two-component mixture with non-reciprocal interaction Consider a two-component mixture of interacting particles in one dimension. The system is described by coupled Dean-Kawasaki equations. Assuming we can linearise them, we obtain for the Fourier-transformed density fluctuations [69, 70] ∂tϕ(1) q(t) = −q2D1ϕ(1) q(t)−q2⟨ρ(1)⟩ γ1ˆ V11(q)ϕ(1) q(t) + ˆ V12(q)ϕ(2) q(t) +iqq2D1⟨ρ(1)⟩ˆ ξ1(q, t),(4.14a) ∂tϕ(2) q(t) = −q2D2ϕ(2) q(t)−q2⟨ρ(2)⟩ γ2ˆ V22(q)ϕ(2) q(t) + ˆ V21(q)ϕ(1) q(t) +iqq2D2⟨ρ(2)⟩ˆ ξ2(q, t),(4.14b) 52 4. Density MBR for macroscopic densities where ˆ Vij(q)is the Fourier transform of the interaction potential between species i and j. We consider T1=T2. But we allow for non-reciprocal interaction between the species, i.e., V12(x−y)=V21(x−y), driving the system out of equilibrium. We can solve Eqs. (4.14) analytically to find the multi-component ISF ˆ S(ij)(q, t), given in Eqs. (B.19) in Ap. B.2.2. For purely non-reciprocal interaction between the two species, i.e., V12(x−y) = −V21(x−y), the diagonal terms Re( ˆ S(11)(q, t)) and Re( ˆ S(22)(q, t)) are the same but Re( ˆ S(12)(q, t)) = Re( ˆ S(21)(q, t)). Looking at the diagonal entries of the dMBR in Fig. 4.2, we find Mρ 11(τ, t, l, q) = Mρ 22(τ, t, l, q)both in and out of equilibrium. For the off-diagonal entries, we see that in equilibrium, Mρ 12(τ, t, l, q) = Mρ 21(τ, t, l, q), and out of equilibrium, Mρ 12(τ, t, l, q)=Mρ 21(τ, t, l, q). Since we know that the ISFs of the two species are the same, we can conclude that DB is broken in the latter case. 0246810 q 2 Dt 0.0 0.1 0.2 0.3 0.4 0.5 ii ( , t , l , q ) 11 22 theory equilibrium nonequilibrium (a) 0246810 q 2 Dt 0.15 0.10 0.05 0.00 0.05 0.10 ij ( , t , l , q ) 12 21 equilibrium nonequilibrium (b) Figure 4.2.: Time evolution of the dMBR for a two-component mixture. The diagonal entries of Mρ ij are shown in (a), the off-diagonal elements in (b) for T1=T2, γ1=γ2, τ = 1/(q2D), l = 0.01, and q=⟨ρ(1)⟩=⟨ρ(2)⟩. In equilibrium, ˆ V12(q) = 1 = ˆ V21(q), while out of equilibrium, ˆ V12(q) = 1 = −ˆ V21(q). However, this method is only applicable to some special cases where the ISFs of different species are the same or known for all time. If we, e.g., consider the twocomponent mixture with different temperatures but reciprocal interaction, we also break DB. But since in this case, ˆ S(11)(q, t)=ˆ S(22)(q, t), we can not use the same method to detect broken DB here. 53 6. Conclusion 6.2. Outlook Some aspects of this thesis might be of interest for future study. While in this thesis, we investigated the parameters for which the deviation of the long-term limit of the dMBR is largest, one might also ask which values of the parameters yield the smallest variance of the dMBR, as done in Ref. [48]. This is especially relevant when evaluating the dMBR from experiments, where minimising the variance can be used to minimise the error bars. Regarding the experiments with the vesicles inside the HeLa cells, more data and longer trajectories would be useful to investigate the long-time limit of the dMBR and to improve statistics. For the entropy estimate in cells, the next step is to estimate the standard error, as was done for the dMBR. This permits us to decide whether a non-zero entropy bound detects a significant amount of entropy, or is just a product of insufficient data. For a better understanding of the vesicle dynamics, a theoretical model yielding qualitatively similar results for the dMBR would be helpful. One candidate is a horse-and-cart model with an additional bath particle as used for a bead in a cell (see Ref. [48]). The trap is used to mimic the cytoskeleton of the cell, which is a candidate to explain the long-range interactions observed between the vesicles. Hence, having several such horse-and-cart systems with interacting traps might be a possible candidate for the vesicles. To also capture the imaginary part of the dMBR, a non-Gaussian coupling between the traps is needed. Another way to model the cytoskeleton is by a connected series of segments with constant velocity to which the vesicles can connect for some time [82]. Alternatively, there exist more complex models for vesicle transport trying to mimic more details of the cell [83]. For the dMBR for macroscopic densities, it remains to find a model where the dMBR can detect broken DB by deviating from 1/2. As we found, this requires inhomogeneity or non-Gaussianity. Also, experimental testing might be an interesting future task. One option here is to look at pictures of, e.g., cells (or other active systems), where the density is proportional to the intensity of a pixel. This way, we easily have access to the density and its Fourier transform. Since cells typically are non-Gaussian and inhomogeneous, we might be able to see broken DB from such experiments. 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This path has to fulfil the differential equation (2.17), which for the system of two harmonically coupled particles reads ¨ y y y−  0k γ1(1 −T1 T2) k γ2(1 −T2 T1) 0  ˙ y y y−  k2 γ2 1+k2 γ1γ2 T1 T2−k2 γ2 1−k2 γ1γ2 T1 T2 −k2 γ2 2−k2 γ1γ2 T2 T1 k2 γ2 2+k2 γ1γ2 T2 T1  y y y= 0.(A.1) This second-order differential equation can be solved analytically. Since we want to know the transition probability P(x x xt, t|x x x0,0), we solve the differential equation with boundary conditions y y y(0) = x x x0,y y y(t) = x x xt. The resulting path of minimum action can be used to calculate the minimum action Smin. Solving the integral in Eq. (2.14) for the path y y y(t), we obtain Smin =− k2t1−e−k γµt−1 (γ1+γ2)(γ1T1+γ2T2)xr,t −e−k γµtxr,02 2kBkt 1 + e−k γµt(γ1T1+γ2T2)(γ2T1+γ1T2)−2γ1γ2γµ(T1−T2)21−e−k γµt −k(γ1+γ2)2(1 + e−k γµt)(γ2T1+γ1T2)(xs,t −xs,0)2 4kBkt 1 + e−k γµt(γ1T1+γ2T2)(γ2T1+γ1T2)−2γ1γ2γµ(T1−T2)21−e−k γµt −γµk(γ1+γ2)2(T2−T1)(xs,t −xs,0)(xr,t −e−2k γµtxr,0) kBkt 1 + e−k γµt(γ1T1+γ2T2)(γ2T1+γ1T2)−2γ1γ2γµ(T1−T2)21−e−k γµt. (A.2) 69 B. Density MBR and ISF B.1. Single component Consider a homogeneous system with Gaussian density fluctuations as discussed in Sec. 4.1.1. To derive Eq. (4.6), we follow the derivation of the MBR-MSD formula in Ref. [7]. For the dMBR, we have to keep in mind that ρq(t)is complex. Therefore, Eqs. (26) and (27) in Ref. [7] for the densities become ⟨ρq(t)−ρq(0)⟩|d= ∂ ∂(ikre)RDρq(˜ t)d2q (2π)2eiqreRe(ρq(0)−ρq(−τ)−d)+iqim Im(ρq(0)−ρq(−τ)−d)eikre Re(ρq(t)−ρq(0))+ikim Im(ρq(t)−ρq(0))e−A[ρq(˜ t)] RDρq(˜ t)d2q (2π)2eiqreRe(ρq(0)−ρq(−τ)−d)+iqim Im(ρq(0)−ρq(−τ)−d)eikre Re(ρq(t)−ρq(0))+ikim Im(ρq(t)−ρq(0))e−A[ρq(˜ t)] k k k=0 + ∂ ∂kim RDρq(˜ t)d2q (2π)2eiqreRe(ρq(0)−ρq(−τ)−d)+iqim Im(ρq(0)−ρq(−τ)−d)eikre Re(ρq(t)−ρq(0))+ikim Im(ρq(t)−ρq(0))e−A[ρq(˜ t)] RDρq(˜ t)d2q (2π)2eiqreRe(ρq(0)−ρq(−τ)−d)+iqim Im(ρq(0)−ρq(−τ)−d)eikre Re(ρq(t)−ρq(0))+ikim Im(ρq(t)−ρq(0))e−A[ρq(˜ t)] k k k=0 (B.1) = ∂ ∂(ikre)Rd2q (2π)2eiqreRe(d)+iqim Im(d)⟨eiqre Re(ρq(0)−ρq(−τ))+iqim Im(ρq(0)−ρq(−τ))eikre Re(ρq(t)−ρq(0))+ikim Im(ρq(t)−ρq(0))⟩ Rd2q (2π)2eiqreRe(d)+iqim Im(d)⟨eiqre Re(ρq(0)−ρq(−τ))+iqim Im(ρq(0)−ρq(−τ))eikre Re(ρq(t)−ρq(0))+ikim Im(ρq(t)−ρq(0))⟩k k k=0 + ∂ ∂kim Rd2q (2π)2eiqreRe(d)+iqim Im(d)⟨eiqre Re(ρq(0)−ρq(−τ))+iqim Im(ρq(0)−ρq(−τ))eikre Re(ρq(t)−ρq(0))+ikim Im(ρq(t)−ρq(0))⟩ Rd2q (2π)2eiqreRe(d)+iqim Im(d)⟨eiqre Re(ρq(0)−ρq(−τ))+iqim Im(ρq(0)−ρq(−τ))eikre Re(ρq(t)−ρq(0))+ikim Im(ρq(t)−ρq(0))⟩k k k=0 , (B.2) where k k k= (kre, kim). The first line in Eq. (B.2) yields the real part of the conditioned mean, and the second line the imaginary part. For a Gaussian process with ⟨ρq(t)− ρq(0)⟩= 0 for all times, we can calculate the expectation values ⟨eiqreRe(ρq(0)−ρq(−τ))+iqimIm(ρq(0)−ρq(−τ))eikreRe(ρq(t)−ρq(0))+ikimIm(ρq(t)−ρq(0))⟩ = exp"−1 2 q2 re⟨Re2(ρq(0) −ρq(−τ))⟩+q2 im⟨Im2(ρq(0) −ρq(−τ))⟩+k2 re⟨Re2(ρq(t)−ρq(0))⟩ +k2 im⟨Im2(ρq(t)−ρq(0))⟩!−qrekre⟨Re(ρq(0) −ρq(−τ))Re(ρq(t)−ρq(0))⟩ −qimkim⟨Im(ρq(0) −ρq(−τ))Im(ρq(t)−ρq(0))⟩#. (B.3) 76 B. Density MBR and ISF Here, we used that the real and the imaginary part of ρq(t)−ρq(t′)are uncorrelated, as is the case for homogeneous systems with Gaussian density fluctuations (see Eq. (4.4)). We can further use that the correlation of the real parts and the correlation of the imaginary parts are the same, and replace all imaginary parts with real parts. Inserting the resulting expression for Eq. (B.3) into Eq. (B.2), we obtain ⟨ρq(t)−ρq(0)⟩|d =Rd2q(iqre)e−iqreRe(d)−iqimIm(d)e−1 2(q2 re+q2 im)⟨Re2(ρq(0)−ρq(−τ))⟩⟨Re(ρq(0) −ρq(−τ))Re(ρq(t)−ρq(0))⟩ Rd2qe−iqreRe(d)−iqimIm(d)e−1 2(q2 re+q2 im)⟨Re2(ρq(0)−ρq(−τ))⟩ +Rd2q(−qim)e−iqreRe(d)−iqimIm(d)e−1 2(q2 re+q2 im)⟨Re2(ρq(0)−ρq(−τ))⟩⟨Re(ρq(0) −ρq(−τ))Re(ρq(t)−ρq(0))⟩ Rd2qe−iqreRe(d)−iqimIm(d)e−1 2(q2 re+q2 im)⟨Re2(ρq(0)−ρq(−τ))⟩. (B.4) After completing the square in the exponentials, we can solve the integral over qre and qim. This results in ⟨ρq(t)−ρq(0)⟩|d=Re(d)⟨Re(ρq(0) −ρq(−τ))Re(ρq(t)−ρq(0))⟩ ⟨Re2(ρq(0) −ρq(−τ))⟩ +iIm(d)⟨Re(ρq(0) −ρq(−τ))Re(ρq(t)−ρq(0))⟩ ⟨Re2(ρq(0) −ρq(−τ))⟩ =d⟨Re(ρq(0) −ρq(−τ))Re(ρq(t)−ρq(0))⟩ ⟨Re2(ρq(0) −ρq(−τ))⟩.(B.5) Finally, we can insert the conditioned mean in Eq. (3.11) to obtain the dMBR. The dependence on dcancels out and we are left with the integral Rddϑl(d)P(d) = 1, such that the dMBR is Mρ(τ, t, l, q) = −⟨Re(ρq(0) −ρq(−τ))Re(ρq(t)−ρq(0))⟩ ⟨Re2(ρq(0) −ρq(−τ))⟩.(B.6) Using again that the correlation functions are the same for real and imaginary parts, we can rewrite Eq. (B.6) as Mρ(τ, t, l, q) = −Re(⟨(ρq(0) −ρq(−τ))(ρq(t)−ρq(0))∗⟩) Re(⟨(ρq(0) −ρq(−τ))(ρq(0) −ρq(−τ))∗⟩).(B.7) Now, we can identify the ISF ˆ S(q, t) = ⟨ρq(t)ρq(0)∗⟩and find Mρ(τ, t, l, q) = 1 2 1−Re ˆ S(q, t +τ)−ˆ S(q, t) Re ˆ S(q, τ)−ˆ S(q, 0) ,(B.8) 77 B. Density MBR and ISF which is identical to Eq. (4.6) in Sec. 4.1.1. B.2. Multiple components B.2.1. Derivation of the dMBR The derivation of Eq. (4.12) is analogous to the derivation in Ap. B.1. The main difference is that now ρq(t)is a vector. The dimension of these vectors is the number of mixture components. We want to calculate the conditioned mean of one component with only one mixture component deviation fixed, i.e. ⟨ρ(i) q(t)−ρ(i) q(0)⟩|ρ(j) q(0)−ρ(j) q(−τ)=d. Therefore, we just have to replace every ρq(0) −ρq(−τ)in the derivation in Ap. B.1 by ρ(j) q(0) −ρ(j) q(−τ), and every ρq(t)−ρq(0) by ρ(i) q(t)−ρ(i) q(0). The calculation is then exactly the same, and Eq. (B.7) becomes Mρ ij(τ, t, l, q) = −Re(⟨(ρ(j) q(0) −ρ(j) q(−τ))(ρ(i) q(t)−ρ(i) q(0))∗⟩) Re(⟨(ρ(j) q(0) −ρ(j) q(−τ))(ρ(j) q(0) −ρ(j) q(−τ))∗⟩).(B.9) Identifying the ISF, we obtain Eq. (4.12). B.2.2. ISF of a two-component mixture Consider a two-component mixture with densities described by the coupled DeanKawasaki equations (4.14). First of all, we realise that this is a linearly coupled system. Therefore, with an adequate variable transformation, we can decouple the equations of motion. In vector notation, Eqs. (4.14) are ∂tϕ ϕ ϕq=−q2  D1+⟨ρ(1)⟩ γ1 ˆ V11(q)⟨ρ(1)⟩ γ1 ˆ V12(q) ⟨ρ(2)⟩ γ2 ˆ V21(q)D2+⟨ρ(2)⟩ γ2 ˆ V22(q) ϕ ϕ ϕq+iq  q2D1⟨ρ(1)⟩0 0q2D2⟨ρ(2)⟩ ξ ξ ξ. (B.10) Here, ξ ξ ξis Gaussian white noise with correlation function Iδ(t−t′).The equations decouple in the basis in which the interaction matrix, i.e., the first matrix in the above equation, is diagonal. To shorten the notation, we introduce   D1+⟨ρ(1)⟩ γ1 ˆ V11(q)⟨ρ(1)⟩ γ1 ˆ V12(q) ⟨ρ(2)⟩ γ2 ˆ V21(q)D2+⟨ρ(2)⟩ γ2 ˆ V22(q) =  a1b c a2 .(B.11) 78 B. Density MBR and ISF This matrix can be brought to diagonal form   a1b c a2 =A A A  λ−0 0λ+ A A A−1,(B.12) with eigenvalues λ−=a1+a2−q(a1+a2)2+ 4bc 2,(B.13a) λ+=a1+a2+q(a1+a2)2+ 4bc 2.(B.13b) The basis transformation matrices are A A A=  −2b a1−a2+√(a1+a2)2+4bc −2b a1−a2−√(a1+a2)2+4bc 1 1   ,(B.14a) A A A−1=c q(a1+a2)2+ 4bc    −1−2b a1−a2−√(a1+a2)2+4bc 12b a1−a2+√(a1+a2)2+4bc   .(B.14b) For the transformed variables ˜ ϕ ˜ ϕ ˜ ϕ=A A A−1ϕ ϕ ϕ, Eq. (B.10) becomes ∂t˜ ϕ ˜ ϕ ˜ ϕ=−q2  λ−0 0λ+ ˜ ϕ ˜ ϕ ˜ ϕ+iqA A A−1 q2D1⟨ρ(1)⟩0 0q2D2⟨ρ(2)⟩ ξ. (B.15) We can solve this equation and find that given an initial condition, the transformed density is ˜ ϕ(1) q(t) =˜ ϕ(1) q(0)e−q2λ−t+c q(a1+a2)2+ 4bciq ×Zt 0dt′e−q2λ−(t−t′)(−q2D1⟨ρ(1)⟩ξ1(t′) + A−1 12 q2D2⟨ρ(2)⟩ξ2(t′)),(B.16a) ˜ ϕ(2) q(t) =˜ ϕ(2) q(0)e−q2λ+t+c q(a1+a2)2+ 4bciq ×Zt 0dt′e−q2λ+(t−t′)(q2D1⟨ρ(1)⟩ξ1(t′) + A−1 22 q2D2⟨ρ(2)⟩ξ2(t′)).(B.16b) 79 B. Density MBR and ISF Here, A−1 ij are the corresponding elements of the matrix A A A. Now, we can calculate the ϕ ϕ ϕ(t) = A A A˜ ϕ ˜ ϕ ˜ ϕ(t)and find ϕ(1) q(t) =A11 ˜ ϕ(1) q(0)e−q2λ−t+A12 ˜ ϕ(2) q(0)e−q2λ+t+c q(a1+a2)2+ 4bciq ×Zt 0dt′e−q2λ−(t−t′)A11(−q2D1⟨ρ(1)⟩ξ1(t′) + A−1 12 q2D2⟨ρ(2)⟩ξ2(t′)) +e−q2λ+(t−t′)A12(q2D1⟨ρ(1)⟩ξ1(t′) + A−1 22 q2D2⟨ρ(2)⟩ξ2(t′)),(B.17a) ϕ(2) q(t) =˜ ϕ(1) q(0)e−q2λ−t+˜ ϕ(2) q(0)e−q2λ+t+c q(a1+a2)2+ 4bciq ×Zt 0dt′e−q2λ−(t−t′)A11(−q2D1⟨ρ(1)⟩ξ1(t′) + A−1 12 q2D2⟨ρ(2)⟩ξ2(t′)) +e−q2λ+(t−t′)A12(q2D1⟨ρ(1)⟩ξ1(t′) + A−1 22 q2D2⟨ρ(2)⟩ξ2(t′)).(B.17b) Knowing the correlation functions of the noise, we can calculate the correlation function ⟨ϕ(1) q(t)ϕ(1)∗ q(t′)⟩=⟨˜ ϕ(1)2 q(0)⟩A2 11e−q2λ−(t+t′)+⟨˜ ϕ(2)2 q(0)⟩A2 12e−q2λ+(t+t′) +A11A12⟨˜ ϕ(1) q(0)˜ ϕ(2) q(0)⟩(e−q2(λ−t+λ+t′)+e−q2(λ−t′+λ+t)) + c2 (a1+a2)2+ 4bc ×"1 λ− e−q2λ−|t−t′|A2 11(D1⟨ρ(1)⟩+A−1 12 D2⟨ρ(1)⟩) + 1 λ+ e−q2λ+|t−t′|A2 12(D1⟨ρ(1)⟩+A−1 22 D2⟨ρ(1)⟩) +2 λ++λ− A11A12(−D1⟨ρ(1)⟩+A−1 12 A−1 22 D2⟨ρ(1)⟩)(e−q2λ−|t−t′|+e−q2λ+|t−t′|)#. (B.18) We are interested in the correlator in the stationary state. Since the system will relax to its stationary state, we obtain the steady state correlator in the limit t, t′→ ∞ while keeping t−t′fixed. We can calculate the other correlators in the same way. 80 B. Density MBR and ISF This leaves us with ⟨ϕ(1) q(t)ϕ(1)∗ q(0)⟩=c2 (a1+a2)2+ 4bc ×"1 λ− e−q2λ−|t−t′|A2 11(D1⟨ρ(1)⟩+A−1 12 D2⟨ρ(1)⟩) + 1 λ+ e−q2λ+|t−t′|A2 12(D1⟨ρ(1)⟩+A−1 22 D2⟨ρ(1)⟩) +2 λ++λ− A11A12(−D1⟨ρ(1)⟩+A−1 12 A−1 22 D2⟨ρ(1)⟩)(e−q2λ−|t−t′|+e−q2λ+|t−t′|)#, (B.19a) ⟨ϕ(2) q(t)ϕ(2)∗ q(0)⟩=c2 (a1+a2)2+ 4bc ×"1 λ− e−q2λ−|t−t′|(D1⟨ρ(1)⟩+A−1 12 D2⟨ρ(1)⟩) + 1 λ+ e−q2λ+|t−t′|(D1⟨ρ(1)⟩+A−1 22 D2⟨ρ(1)⟩) +2 λ++λ− (−D1⟨ρ(1)⟩+A−1 12 A−1 22 D2⟨ρ(1)⟩)(e−q2λ−|t−t′|+e−q2λ+|t−t′|)#,(B.19b) ⟨ϕ(1) q(t)ϕ(2)∗ q(0)⟩=−c2 (a1+a2)2+ 4bc ×"1 λ− e−q2λ−|t−t′|A11(D1⟨ρ(1)⟩+A−1 12 D2⟨ρ(1)⟩) + 1 λ+ e−q2λ+|t−t′|A12(D1⟨ρ(1)⟩+A−1 22 D2⟨ρ(1)⟩) +2 λ++λ− (−D1⟨ρ(1)⟩+A−1 12 A−1 22 D2⟨ρ(1)⟩)(A11e−q2λ−|t−t′|+A12e−q2λ+|t−t′|)#, (B.19c) ⟨ϕ(1) q(t)ϕ(2)∗ q(0)⟩=−c2 (a1+a2)2+ 4bc ×"1 λ− e−q2λ−|t−t′|A11(D1⟨ρ(1)⟩+A−1 12 D2⟨ρ(1)⟩) + 1 λ+ e−q2λ+|t−t′|A12(D1⟨ρ(1)⟩+A−1 22 D2⟨ρ(1)⟩) +2 λ++λ− (−D1⟨ρ(1)⟩+A−1 12 A−1 22 D2⟨ρ(1)⟩)(A12e−q2λ−|t−t′|+A11e−q2λ+|t−t′|)#. (B.19d) To obtain the ISF, i.e., ⟨ρq(t)ρ∗ q(0), we remind ourselves of the definition ϕ= (ρ− ⟨ρ⟩)/q⟨ρ⟩. This means, ρq=ϕqq⟨ρ⟩+⟨ρq⟩. Since we are in a homogeneous state, ⟨ρq⟩= 0 for all finite q. Therefore, the ISF and the correlation functions (B.19) only differ by a factor. Calculating the dMBR using Eq. (4.12), this factor cancels out, such that we can insert Eqs. (B.19) into Eq. (4.12) instead of the ISF. 81 Acknowledgement First and foremost, I would like to thank Matthias Krüger for supervising my thesis, answering my questions and providing many helpful ideas. My thanks also go to Timo Betz for agreeing to be the second referee, as well as for providing practical input during our weekly meetings. Furthermore, I want to thank the entire RG Krüger for many discussions in and outside the Group meeting. A special thanks goes to Iustus Hemprich, not only for the useful discussion but also for bringing plants and pleasant distractions to our office, and for proofreading. I am also grateful to Sarah Lädke for bringing in the experimentalist point of view in our discussions and for the great collaboration regarding the vesicle project, as well as for proofreading. Thanks a lot to Anne, Lars, Elke, Julia and Annemarie for their support, not only in proofreading. 82 Erklärung Ich versichere hiermit, dass ich die vorliegende Arbeit ohne fremde Hilfe selbstständig verfasst und nur die von mir angegebenen Quellen und Hilfsmittel verwendet habe. Wörtlich oder sinngemäß aus anderen Werken entnommene Stellen habe ich unter Angabe der Quellen kenntlich gemacht. Die Richtlinien zur Sicherung der guten wissenschaftlichen Praxis an der Universität Göttingen wurden von mir beachtet. Eine gegebenenfalls eingereichte digitale Version stimmt mit der schriftlichen Fassung überein. Mir ist bewusst, dass bei Verstoß gegen diese Grundsätze die Prüfung mit nicht bestanden bewertet wird. Göttingen, den November 11, 2025 (Laila Henkes)