Macroscopic Fluctuation–Dissipation Relations Away from Equilibrium

Bachelor’s Thesis Macroscopic Fluctuation–Dissipation Relations Away from Equilibrium Makroskopische Fluktuations-Dissipations-Beziehungen außerhalb des Gleichgewichts prepared by Julius Lüning from Bremen at the Institut für Theoretische Physik of Georg-August-Universität Göttingen Thesis period: 5th August 2025 until 11th November 2025 First referee: Prof. Dr. Matthias Krüger Second referee: Prof. Dr. Peter Sollich abstract The fluctuation–dissipation theorem (FDT) establishes a fundamental connection between a system’s equilibrium fluctuations and its linear response to external perturbations. While valid near equilibrium, it breaks down under strong perturbations, when the system’s response becomes nonlinear. In this thesis, we extend the theoretical framework describing nonequilibrium situations beyond the FDT. In particular, we investigate a system driven far away from equilibrium by a multidimensional time-dependent protocol, which generalizes earlier results on one-dimensional driving. To analyze the system’s nonequilibrium behavior, we introduce generalized connected correlation functions (GCCFs), which we identify as tensor-like quantities capturing the system’s statistical properties. Using nonlinear response relations derived within the path integral formalism, the components of the GCCFs are expressed as Volterra series expansions around the reference equilibrium state of the system. Microscopic expressions for the memory kernel components, arising from these expansions, are derived and shown to be connected through a set of component-wise nonlinear fluctuation–dissipation relations. These include a component-wise FDT as the lowest-order relation. From the expansions we derive a response relation that quantifies the violation of the FDT under multidimensional driving. Finally, we demonstrate that the one-dimensional driving case is naturally recovered within our framework and evaluate the FDT violation for a constant velocity protocol with two components. iii Contents 1. Introduction 1 2. Background 5 2.1. Linear Response Theory . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2. SystemSetup ............................... 7 2.3. Nonlinear Response Theory within Path Integral Formalism . . . . . 7 2.3.1. Nonlinear Response via Equilibrium Averages . . . . . . . . . 8 2.3.2. Nonlinear Response via Equilibrium Joint Cumulants . . . . . 11 2.4. Volterra Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . 12 3. Multidimensional Driving Protocols 13 3.1. Microscopic Action Components . . . . . . . . . . . . . . . . . . . . . 14 3.2. Generalized Connected Correlation Functions (GCCFs) . . . . . . . . 17 3.3. Nonlinear Volterra Series Expansion . . . . . . . . . . . . . . . . . . . 19 3.3.1. Formulation of the Expansions . . . . . . . . . . . . . . . . . . 19 3.3.2. Microscopic Expressions for the Kernel Components . . . . . . 22 3.4. Fluctuation-Dissipation Relations . . . . . . . . . . . . . . . . . . . . 25 3.5. Violation of the Fluctuation-Dissipation Theorem . . . . . . . . . . . 27 4. Applications of the Formalism 30 4.1. One-Dimensional Driving . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2. Two-Component Velocity Protocol . . . . . . . . . . . . . . . . . . . 32 5. Summary and Outlook 36 A. Joint Cumulants and Modified Response Relations 40 A.1. Definition of Joint Cumulants . . . . . . . . . . . . . . . . . . . . . . 40 A.2. Derivation of the Nonlinear Response via Equilibrium Joint Cumulants 41 v Contents B. Derivation of the Volterra Series Expansions 44 B.1. Expansion of the Mean . . . . . . . . . . . . . . . . . . . . . . . . . . 44 B.2. Expansion of the Covariance Components . . . . . . . . . . . . . . . 48 C. Validity of the Fluctuation-Dissipation Relations 52 Acknowledgments 59 vi 1. Introduction Most physical systems in the real world consist of a vast number of microscopic constituents. Their macroscopic properties emerge from the collective behavior of these components, with microscopic fluctuations often playing a crucial role. One of the most fundamental everyday examples is the concept of temperature in a many-body system. On the microscopic scale, the particles are in constant motion, each with a fluctuating kinetic energy. Macroscopically on the other hand, these fluctuations average out, and the mean kinetic energy per particle becomes timeindependent. Temperature is then defined as a quantity proportional to this average, making it a macroscopic property that originates from the microscopic dynamics [1]. Yet perhaps the most famous phenomenon that can be understood in this way is Brownian motion. Discovered by the Scottish botanist Robert Brown in 1827 [2], it describes the erratic, random motion of a colloidal particle1, often referred to as a Brownian particle, when immersed in a fluid. This motion is visible under an optical microscope and arises from countless microscopic collisions of the Brownian particle with the surrounding, much faster-moving solvent molecules [4]. These examples illustrate the importance of establishing a theoretical framework to quantify microscopic behavior and connect it to observable macroscopic properties. A crucial aspect such a framework needs to consider, is the fundamental distinction between equilibrium situations, where thermodynamic observables are time-independent [5], and nonequilibrium situations, which describe most systems found in nature [6]. The latter evolve in time and are characterized by irreversible fluxes of matter and energy, as well as entropy production [6,7]. This renders thermodynamic observables time-dependent and makes the theoretical description considerably more difficult. Historically, the theory of Brownian motion has proven to be one of the simplest, yet most effective approaches to describe the dynamics of stochastic systems in and out of equilibrium [4]. Particularly in Newtonian fluids like water, where the vis1Colloids are particles whose sizes typically range from about 1 nm to 1 µm [3]. 1 1. Introduction cosity is independent of the shear rate [8], Brownian motion is well studied. In such fluids, the Brownian particle relaxes on much longer time scales than the surrounding solvent bath molecules [9]. As a result, the solvent bath can be considered memoryless2or Markovian. The equation of motion describing the temporal evolution of the Brownian particle is the Langevin equation. It consists of a deterministic part, arising from friction and external forces, and a stochastic contribution that represents the random collisions with the surrounding solvent molecules [4,5]. While such systems have been extensively studied both in equilibrium and in nonequilibrium situations, the setting of more complex solvent baths remains much less understood. Many biologically interesting fluids, such as nucleic acid condensates (e.g., DNA or RNA) [10], cytoplasm [11], or even cerebrospinal fluid3[12], fall into this category. All of them can be classified as non-Newtonian viscoelastic fluids [10–13], characterized by long relaxation times that are of the same order of magnitude or longer than those of Brownian particles [9,13]. This results in memory effects, where the system’s future evolution depends on its past history, not just the present state. Therefore, such baths can no longer be considered Markovian, and we require a modified non-Markovian description of the system [4]. Especially in nonequilibrium situations, such as when a Brownian particle immersed in a nonlinear, non-Markovian bath is strongly driven, the theoretical description becomes challenging due to the complex interactions between the driven particle and the nonlinear environment [14]. When the system is weakly driven and remains close to equilibrium, its response is linear and well described by the fluctuation–dissipation theorem (FDT), which establishes a connection between the linear response to small external perturbations and the system’s internal equilibrium fluctuations [15]. However, under stronger driving, the system exhibits a nonlinear response and nonlinear fluctuations [14,16] that are no longer captured by the FDT, requiring an extended theoretical framework. Such nonequilibrium behavior has been observed in several instances, including flow instabilities in experiments with worm-like micellar fluids [17] and shear-thinning or shear-thickening behavior in active baths [18]. Over the years, various methods have been developed to investigate nonequilibrium systems. One of the earliest was introduced in the 1950s by Ryogo Kubo, ex2Memoryless means that the future evolution of the system depends only on its present state, not on its past history [4,5]. 3This is the protective fluid that circulates through the brain and spinal cord, supporting the central nervous system in several ways [12]. 2 2. Background parts of the action yields S(θχ, θω) = −S(χ, ω)and D(θχ, θω) = D(χ, ω),(2.12) which motivates the terminology. By using that the system was prepared in equilibrium at time t0and the invariance of the equilibrium distribution under time reversal, Peq(xt0, θω) = Peq(xt0, ω), we can express the time-antisymmetric component from Eq. (2.10) as the logarithm of the ratio of path weights: S=A(θχ, θω)− A(χ, ω) = ln e−A(χ,ω)Peq(xt0, ω) e−A(θχ,θω)Peq(xt0, θω)= ln P(χ, ω) P(θχ, θω).(2.13) Expanding the action in Eq. (2.10) yields A=A′+A′′ 2+A′′′ 6+. . . =D′−S′ 2+1 2D′′ −S′′ 2+1 6D′′′ −S′′′ 2+. . . (2.14) To recover the equilibrium path weight Peq(xt0, ω)for a vanishing perturbation, we assume A(0) =D(0) −S(0) 2= 0. Inserting Eq. (2.14) into the perturbed path weight, Eq. (2.9), and expanding the exponential gives P(χ, ω) = 1− A′−1 2A′′ −(A′)2−1 6A′′′ −3A′A′′ + (A′)3+. . . Peq(xt0, ω) = 1−D′−S′ 2!−1 2"D′′ −S′′ 2−D′−S′ 22# −1 6"D′′′ −S′′′ 2−3D′−S′ 2D′′ −S′′ 2+D′−S′ 23# +. . .  Peq(xt0, ω). (2.15) 9 2. Background Using this expression, Eq. (2.8) evaluates to ⟨O⟩=⟨O⟩eq −D′−S′ 2Oeq −1 2"D′′ −S′′ 2Oeq −D′−S′ 22 Oeq# −1 6"D′′′ −S′′′ 2)Oeq −3D′−S′ 2D′′ −S′′ 2Oeq +D′−S′ 23 Oeq#+. . . (2.16) Eq. (2.16) is the nonlinear response of any path observable O. We observe that its expressed in terms of equilibrium averages2⟨F(χ, ω)⟩eq =RDωPeq(xt0, ω)F(χ, ω), where the protocol is fixed at xt0. To derive an analogous relation for the time-reversed observable Oθ, we need to consider the temporal symmetry of the action components, Eq. (2.12), and the already mentioned reversibility of the equilibrium distribution. The latter directly implies that the equilibrium state is invariant under time reversal, i.e., ⟨F(θχ, θω)⟩eq =⟨F(χ, ω)⟩eq holds. The nonlinear response relation for the timereversed path observable then reads ⟨Oθ⟩=⟨O⟩eq −D′+S′ 2Oeq −1 2"D′′ +S′′ 2Oeq −D′+S′ 22 Oeq# −1 6"D′′′ +S′′′ 2Oeq −3D′+S′ 2D′′ +S′′ 2Oeq +D′+S′ 23 Oeq#+. . . (2.17) Once again, the response is entirely expressed in terms of equilibrium averages. Instead of a path observable, we can consider the special case of a state observable Ot(χ, ω) = O(xt,yt), which only depends on the end of the path. To obtain a corresponding nonlinear response relation, we evaluate the difference between Eq. (2.16) and Eq. (2.17) for O=Ot. Due to being only dependent on the end of the path, we define the expectation value of the time-reversed state observable as [16] ⟨(Oθ)t⟩ ≡ ZDω e−A(θχ,θω)Peq(xt0, θω)O(xt,yt).(2.18) By applying the definition of the time reversal operator (2.11) and using the prop2F(χ, ω)serves as an example of another arbitrary path observable. 10 2. Background erties of equilibrium, we find ⟨(Oθ)t⟩=⟨Ot0⟩=⟨Ot0⟩eq =⟨Ot⟩eq.(2.19) The nonlinear response of the state observable Otis then given by ⟨Ot−(Oθ)t⟩=⟨Ot⟩−⟨Ot⟩eq =⟨S′Ot⟩eq +1 2h⟨S′′Ot⟩eq −2⟨D′S′Ot⟩eqi+1 6"⟨S′′′Ot⟩eq −3⟨(D′S′′ +D′′S′)Ot⟩eq +3(D′)2S′+(S′)3 4Oteq#+. . . (2.20) 2.3.2. Nonlinear Response via Equilibrium Joint Cumulants As we have seen, the nonlinear responses of both path observables and state observables, Eqs. (2.16) and (2.20), are entirely expressed in terms of equilibrium averages. In some special application cases, it has been observed that for large times tthe terms in Eq. (2.20) diverge independently [14]. Analytically, this can be avoided by reformulating the response in terms of equilibrium joint cumulants ⟨·;. . . ;·⟩eq, which we refer to as connected correlation functions (CCFs) [14,16,24]. This approach can be applied to both state and path observables. The mathematical definition of joint cumulants, along with the derivation steps leading to the modified response relations, are provided in Appendix A. The path response ⟨O⟩=⟨O(χ, ω)⟩and the state response ⟨Ot⟩=⟨O(xt,yt)⟩, in terms of equilibrium joint cumulants, are given by ⟨O⟩=⟨O⟩eq +1 2⟨S′;O⟩eq − ⟨D′;O⟩eq +1 2h⟨D′;D′;O⟩eq − ⟨D′;S′;O⟩eqi −1 2⟨D′′;O⟩eq +1 4⟨S′′;O⟩eq +1 8⟨S′;S′;O⟩eq +. . . , (2.21) ⟨Ot⟩=⟨Ot⟩eq +⟨S′;Ot⟩eq +1 2⟨S′′;Ot⟩eq − ⟨D′;S′;Ot⟩eq −1 2h⟨D′;S′′;Ot⟩eq +⟨D′′;S′;Ot⟩eqi+1 2⟨D′;D′;S′;Ot⟩eq +1 24⟨S′;S′;S′;Ot⟩eq +1 6⟨S′′′;Ot⟩eq +. . . (2.22) In the following, we will work with these modified response relations. 11 2. Background 2.4. Volterra Series Expansion We will see that the nonlinear response and fluctuations under strong external driving, obtained from Eqs. (2.21) and (2.22), can be cast into a functional expansion around the reference equilibrium state of the system. This expansion does not take the simple form of a Taylor series, due to the time-dependent protocol. Instead, it must account for the memory effects, and is known as a Volterra series expansion. In this subsection we want to briefly introduce the mathematical formulation. Formally, the Volterra series is a functional expansion that quantifies the response of a nonlinear system with memory to an arbitrary input [28,29]. The key difference from a Taylor series is that the system’s response at a given time depends on the input at all previous times, rather than only on the input at the current time. For a causal3and time-varying system, the expansion is given by [29] y(t) = h0(t) + Zt t0 dτ1h1(t, τ1)x(τ1) + Zt t0 dτ1Zt t0 dτ2h2(t, τ1, τ2)x(τ1)x(τ2) +Zt t0 dτ1Zt t0 dτ2Zt t0 dτ3h3(t, τ1, τ2, τ3)x(τ1)x(τ2)x(τ3) + . . . , (2.23) where yis the output, representing the nonlinear response to the input xover the time interval [t0, t]. This expansion gives rise to functions h0(t), h1(t, τ1), h2(t, τ1, τ2), . . . , which are named Volterra kernels (or memory kernels) and characterize how past inputs contribute to the system’s response [29]. The form of the Volterra series used here reflects that the system dynamics vary in time, making the kernels explicitly dependent not only on the integration times τ1, τ2, τ3, . . . , but also on the observation time t. The representation in Eq. (2.23) is unique only if the Volterra kernels are symmetric in the integration variables for each fixed observation time t[29]. To enforce this property, the n-th order kernel hn(t, τ1, . . . , τn)can always be replaced by its symmetrized form [29] hsym n(t, τ1, . . . , τn)≡1 n!X π∈Sn hn(t, τπ(1), . . . , τπ(n)),(2.24) where the sum runs over all elements πin the symmetric group Sn, i.e., the group of all permutations of nelements. This symmetrization leaves the expansion in Eq. (2.23) unchanged. 3A system is considered causal, when its output does not depend on future inputs. 12 3. Multidimensional Driving Protocols To study the system introduced in Sec. 2.2, we introduce the observable conjugate to the driving protocol. We define it as the gradient of the potential Uwith respect to xs∈Rd, Fs≡F(ys,xs)≡ ∇xsU(ys,xs)∈Rd,(3.1) with i-th component Fi s≡∂U(ys,xs) ∂xi s for i= 1, . . . , d. (3.2) The physical interpretation of Fsdepends on the specific form of Uand thus on the nature of the applied perturbation. In the special case where the control parameter components xi srepresent spatial coordinates, Fsdenotes (minus) the force acting on the multidimensional control parameter xs. From this point onward, we will refer to it as the generalized force. When the system is driven far away from equilibrium, its statistical properties are described by the quantities ⟨Bt;Ft1;. . . ;Ftm⟩, which we name generalized connected correlation functions (GCCFs). They depend on another arbitrary state observable Bt=B(yt,xt)and the generalized force F˜ t=F(y˜ t,x˜ t)at times ˜ t=t1, . . . , tm. Since the generalized force is a vector-valued observable, the GCCFs need to be mathematically distinct from the joint cumulants introduced in Sec. 2.3.2 and Appendix A. This work aims to explicitly define these functions and show that they can be represented as nonlinear Volterra series expansions around the system’s reference equilibrium state. 13 3. Multidimensional Driving Protocols 3.1. Microscopic Action Components To apply the response relations Eqs. (2.21) and (2.22), we need to further clarify the two action components Sand D. A basic assumption we make, is that the system satisfies the condition of local detailed balance. One physical formulation of this condition is that the time-antisymmetric part of the perturbation action, as given in Eq. (2.13), is equal to the entropy change of the bath relative to the reference equilibrium state [30]. Hence, Sis also referred to as the entropic component of the action [27]. In our case, this change originates from the multidimensional protocol that drives the system out of equilibrium. Analogous to Refs. [16,24], a simple expression for the entropy change can be obtained by analyzing the system’s thermodynamics. We start from the time-dependent Hamiltonian H(ys,xs,xt0) = U(ys,xt0) | {z } ≡H0(ys,xt0) +U(ys,xs)−U(ys,xt0) | {z } ≡H1(ys,xs,xt0) ,(3.3) describing the total energy of the system at time s. We separate the Hamiltonian into two contributions, H0and H1. The term H0(ys,xt0)corresponds to the equilibrium part of the energy, evaluated in the unperturbed ensemble for the fixed control parameter value xt0. The second contribution H1(ys,xs,xt0)is the perturbation Hamiltonian, which quantifies the energy change of the system relative to the equilibrium state. The total energy change over the entire driving period [t0, t]reads ∆H=H(ys,xs,xt0)|t t0. However, the condition of local detailed balance establishes only a connection between the time-antisymmetric action part and the entropy change connected to the perturbation-dependent part of the energy [30]. Therefore, we only need to evaluate the change of the perturbation Hamiltonian ∆H1=H1(ys,xs,xt0)|t t0. On the other hand, ∆H1can be obtained from the first law of thermodynamics [1] ∆H1=W+Q, (3.4) where Qis the energy transferred from the bath to the system in form of heat and Wis the work applied through the external protocol. Combining both approaches 14 3. Multidimensional Driving Protocols yields ∆H1=H1(ys,xs,xt0)|t t0=W+Q. (3.5) For the work applied we find W=Zt t0 ds˙ xs· ∇xsU=Zt t0 ds˙ xs·F(ys,xs) = Zt t0 ds d X i=1 ˙xi sFi(ys,xs) =Zt t0 ds˙xi sFi(ys,xs),(3.6) where ˙ xs=d dt xs∈Rddenotes the protocol velocity. In Eq. (3.6) we introduced the Einstein summation convention [31] to simplify the notation of the appearing dot product1in the integrand. By inserting Eq. (3.6) into Eq. (3.5), the entropy change S=−βQ (with β= 1/(kBT)) can be written as S=β(−∆H1+W) = βU(yt,xt0)−U(yt,xt) + Zt t0 ds˙xi sFi(ys,xs).(3.7) In theory, we could now write down the desired expansion of S. However, this remains challenging because of the two potential boundary terms in Eq. (3.7). We can simplify the expansion by preparing the system in equilibrium in the infinite past, i.e., t0→ −∞ and by choosing the final value of the control parameter equal to its initial value, xt0=xt. With this choice we do not influence the system at time tand we can perform the expansion around the equilibrium state corresponding to xt[16,23,24]. Then Eq. (3.7) simplifies to S=βZt −∞ ds˙xi sFi(ys,xs)(2.13) = ln P(χ, ω) P(θχ, θω).(3.8) The last equality once again emphasizes the assumption of local detailed balance, which establishes the connection between the system’s thermodynamics and the perturbation action from the path integral formalism. The expansion of Eq. (3.8) is still complicated, since the components of the generalized force Fi s=Fi(ys,xs)all explicitly depend on the multidimensional control parameter. This leads to a nonlinear dependence of Son the driving velocity components ˙xi s, which can be seen by expanding Fi saround the equilibrium state, i.e., 1This is the standard dot product on Rd, because both ˙ xs,Fs∈Rd. 15 3. Multidimensional Driving Protocols in powers of (xi s−xi t0). Using our simplification xi t0=xi t, this can equivalently be written as an expansion in powers of (xi s−xi t). We find Fi(ys,xs) = Fi(ys,xt) + ∂Fi(ys,xs) ∂xj sxs=xt (xj s−xj t) + 1 2 ∂2Fi(ys,xs) ∂xj s∂xk sxs=xt ×(xj s−xj t)(xk s−xk t) + . . . (3.9) =Fi(ys,xt)−Zt −∞ ds′˙xj s′Θ(s′−s)∂Fi(ys,xs) ∂xj sxs=xt +Zt −∞ ds′Zt −∞ ds′′ ˙xj s′˙xk s′′ Θ(s′−s)Θ(s′′ −s)1 2 ∂2Fi(ys,xs) ∂xj s∂xk sxs=xt +. . . (3.10) To display the nonlinearity in the driving velocity we used the identity2xj s−xj t= −Rt −∞ ds′˙xj s′Θ(s′−s)iteratively, which is the component-wise generalization of the relation used in the scalar protocol case [16,24]. Inserting the expanded generalized force components from Eq. (3.10) into Eq. (3.8) yields S=βZt −∞ ds˙xi sFi(ys,xt) | {z } ≡S′ −2βZt −∞ dsZt −∞ ds′˙xi s˙xj s′ ∂Fi(ys,xs) ∂xj sxs=xt Θ(s′−s) | {z } ≡S′′ ·1 2 + 3βZt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ ∂2Fi(ys,xs) ∂xj s∂xk sxs=xt Θ(s′−s)Θ(s′′ −s) | {z } ≡S′′′ ·1 6 +. . . (3.11) =S′+S′′ 2+S′′′ 6+. . . (3.12) For clarity, we once again repeat the defined microscopic expansion coefficients: S′=βZt −∞ ds˙xi sFi(ys,xt)(3.13) S′′ =−2βZt −∞ dsZt −∞ ds′˙xi s˙xj s′ ∂Fi(ys,xs) ∂xj sxs=xt Θ(s′−s)(3.14) S′′′ = 3βZt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ ∂2Fi(ys,xs) ∂xj s∂xk sxs=xt Θ(s′−s)Θ(s′′ −s) (3.15) 2Θ(t) = (0, t < 0 1, t ≥0denotes the Heaviside function. 16 3. Multidimensional Driving Protocols . . . Since all generalized force components and their partial derivatives are evaluated at the fixed control parameter value xt, we redefine3Fi s≡Fi(ys,xs)|xs=xtto simplify the notation in the future. The time-symmetric part of the action Ddepends on the dynamics of the system. Hence, without a more detailed specification of the system, its exact form cannot be stated. Since it is not related to thermodynamics, its also named non-thermodynamic or frenetic component [27]. To match the form of the time-antisymmetric part, we assume a general notation D=D′+D′′ 2+. . . with expansion coefficients D′=Zt −∞ ds˙xi sDi s(3.16) D′′ =Zt −∞ dsZt −∞ ds′˙xi s˙xj s′Di,j s,s′(3.17) . . . The coefficients Di sand Di,j s,s′are unknown coefficients, which can only be determined when examining a specific system. For our purpose we do not need to specify them any further. In general, the expansions of Sand Dcan be continued to arbitrary higher order. Here, we restrict ourselves to the coefficients required to evaluate the response relations in Eqs. (2.21) and (2.22) up to the given order. By extending the response relations to higher order, one would also require higher-order expansion coefficients of the action parts. 3.2. Generalized Connected Correlation Functions (GCCFs) The GCCFs are mathematically distinct from the CCFs4, which were the relevant quantities in the one-dimensional driving case [16,24]. Since Fsis vector-valued and Btis scalar, these functions need to take the form of higher-dimensional objects that capture the properties of all ddriving components. To establish a general definition of the GCCF ⟨Bt;Ft1;. . . ;Ftm⟩of order m+ 1, containing the generalized force m 3Note that with this definition the meaning of Fi shas changed. 4Recall that joint cumulants and connected correlation functions (CCFs) are synonymous. 17 3. Multidimensional Driving Protocols times, we first specify the lowest order cases m= 0,1,2,3explicitly. We begin with the first order (m= 0), which is simply the expectation value of Bt, ⟨Bt⟩ ∈ R.(3.18) Since it does not contain the generalized force, it is a scalar quantity. Therefore, the first-order GCCF coincides with the first-order CCF for a scalar protocol, which is the only special case of equivalence. The second-order GCCF (m= 1) already must have a different structure because it contains the generalized force. It is defined as the vector ⟨Bt;Ft′⟩ ≡      ⟨Bt;F1 t′⟩ . . . ⟨Bt;Fd t′⟩      ∈Rd(3.19) with components (⟨Bt;Ft′⟩)α≡ ⟨Bt;Fα t′⟩. It contains all dcovariances between Btand the components of the generalized force and can thus be interpreted as a covariance vector. Continuing, we define the third order (m= 2) as the matrix ⟨Bt;Ft′;Ft′′ ⟩ ≡         ⟨Bt;F1 t′;F1 t′′ ⟩ ⟨Bt;F1 t′;F2 t′′ ⟩. . . ⟨Bt;F1 t′;Fd t′′ ⟩ ⟨Bt;F2 t′;F1 t′′ ⟩ ⟨Bt;F2 t′;F2 t′′ ⟩. . . ⟨Bt;F2 t′;Fd t′′ ⟩ . . .. . ..... . . ⟨Bt;Fd t′;F1 t′′ ⟩ ⟨Bt;Fd t′;F2 t′′ ⟩. . . ⟨Bt;Fd t′;Fd t′′ ⟩         ∈Rd×d(3.20) with components (⟨Bt;Ft′;Ft′′ ⟩)αβ ≡ ⟨Bt;Fα t′;Fβ t′′ ⟩. This d×dmatrix contains all d2three-point correlators (joint cumulants) between Btand the components of the generalized force. In line with this, the fourth order (m= 3) must be a three-index object ⟨Bt;Ft′;Ft′′ ;Ft′′′ ⟩ ∈ Rd×d×d,(3.21) which contains all d3four-point correlators ⟨Bt;Ft′;Ft′′ ;Ft′′′ ⟩αβγ ≡ ⟨Bt;Fα t′;Fβ t′′ ;Fγ t′′′ ⟩. These expressions show an emerging pattern, which allows us to state the general 18 3. Multidimensional Driving Protocols the ordered pairs (sτ, iτ)and symmetric in the ordered pairs (tν, αν). The symmetry properties for all m, n and any π∈Sn,σ∈Smcan be summarized as Γ(m,n) t−s1,...,t−sn;t−t1,...,t−tmα1,...,αm i1,...,in =Γ(m,n) t−sπ(1),...,t−sπ(n);t−tσ(1),...,t−tσ(m)ασ(1),...,ασ(m) iπ(1),...,iπ(n) . (3.38) In the microscopic expressions for the kernel components, we explicitly enforced symmetry with respect to permutations of the ordered pairs (sτ, iτ)by introducing the sums described above. Lastly, we want to briefly explain why the expansions in Eqs. (3.22)—(3.25) are in fact Volterra series. The system’s nonlinear mean response is given by Eq. (3.22). If the sum in the expansion is written out explicitly, the expression matches the form of the Volterra series presented in Eq. (2.23). The only differences are that we have a multidimensional input and that the kernels turn out to depend only on time differences between the observation time tand the integration times. This follows from the properties of equilibrium and therefore has a physical origin. Consequently, the symmetry properties found for the corresponding kernel components are consistent with the mathematical prediction from Eq. (2.24), modified for time differences and multidimensional inputs. The higher-order expansions in Eqs. (3.23)—(3.25), which describe the nonlinear and non-Gaussian fluctuations of the system, are also Volterra series. This can be seen by extending Eq. (2.23) to the case of multidimensional inputs and outputs. 3.4. Fluctuation-Dissipation Relations The memory kernel components in Eqs. (3.27)–(3.30) and (3.32)–(3.37) are found to be related by a set of identities. They read Γ(0,1) tα=Γ(1,0) tαfor α= 1, . . . , d (3.39) Γ(0,2) s1,s2αβ =1 2X π∈S2Γ(1,1) sπ(1);sπ(2) απ(2) απ(1) −1 2Γ(2,0) s1,s2αβ for α, β = 1, . . . , d (3.40) Γ(0,3) s1,s2,s3αβγ =1 6X π∈S3Γ(1,2) sπ(1),sπ(2);sπ(3) απ(3) απ(1)απ(2) −1 12 X π∈S3Γ(2,1) sπ(1);sπ(2),sπ(3) απ(2)απ(3) απ(1) +1 6Γ(3,0) s1,s2,s3αβγ for α, β, γ = 1, . . . , d. (3.41) . . . 25 3. Multidimensional Driving Protocols These identities connect the memory kernel components appearing in the expansion of the mean, Eq. (3.22), with kernel components from the higher order expansions in Eqs. (3.23)–(3.25). They generalize the identities presented in Refs. [16,24] to the case of multidimensional driving and are valid only if the system obeys local detailed balance. Their validity can be explicitly verified by inserting the microscopic kernel expressions, which is done in Appendix C. In formulating the identities, we have symmetrized them with respect to interchanges of the complete ordered pairs (sτ, iτ)and (tν, αν). This symmetrization is required for the relations to hold mathematically. The lowest-order identity in Eq. (3.39) relates the α-th component of the linear response of the mean (Γ(0,1))αto the equilibrium covariance (zeroth-order response) (Γ(1,0))αof Band the generalized force component Fα. For a fixed α, this statement matches the one from the fluctuation–dissipation theorem in Eq. (2.6), i.e., the linear nonequilibrium response of the mean is governed by the fluctuations in equilibrium. Thus, the identity can be understood as a component-wise fluctuation–dissipation theorem, valid independently for each α= 1, . . . , d and is therefore linear. The second identity in Eq. (3.40) relates the αβ-th component of the secondorder response of the mean (Γ(0,2))αβ to the αβ-th component of the linear response of the covariance components (Γ(1,1))β αand the αβ-th component of the zeroth-order response of the third order GCCF components (Γ(2,0))αβ. The relation is nonlinear, due to connecting non-Gaussian fluctuations with nonlinear response. The third identity in Eq. (3.41) relates the αβγ-th component of the third-order response of the mean (Γ(0,3))αβγ to the αβγ-th component of the second-order response of the covariance components (Γ(1,2))γ αβ, the αβγ-th component of the first-order response of the third order GCCF components (Γ(2,1))βγ αand the αβγ-th component of the zeroth-order response of the fourth order GCCF components (Γ(3,0))αβγ. Hence, this relation is also nonlinear. In summary, the identities in Eqs. (3.39)—(3.41) establish relations between the components of the tensor-like memory kernels that arise from the nonlinear Volterra series expansions. Once again, we emphasize that they were derived under the assumption of local detailed balance. We observe that each identity connects kernel components with the same total sum m+n. For instance, the component-wise FDT, Eq. (3.39) connects components of vector kernels with m+n= 1, corresponding to a linear response relation. The second-order identity in Eq. (3.40) connects components of matrix kernels with m+n= 2, and the third-order identity in Eq. (3.41) 26 3. Multidimensional Driving Protocols connects components of rank-three kernels with m+n= 3. While the first identity is a linear relation corresponding to a component-wise fluctuation–dissipation theorem, the higher-order identities, Eqs. (3.40)–(3.41), represent component-wise nonlinear fluctuation–dissipation relations that extend the component-wise FDT to regimes of stronger multidimensional driving. Both of the latter are symmetrized in the way described above. We expect that the list of identities continues to higher order, although these have not yet been calculated. To derive these, an extension of the Volterra series expansions to higher orders would be required. As already noted, these identities can be interpreted as a generalization of those presented in Refs. [16,24], which apply to the case of a scalar driving protocol. Later on we will show, that these identities are included in the ones derived here. We expect the component-wise nonlinear fluctuation–dissipation relations to hold on both microscopic and macroscopic length scales, since neither the observable Bnor the generalized force components Fαhave been specified. The form of the latter depends on the potential energy U. Consequently, the components Fαmay represent either microscopic or macroscopic observables, depending on the applied perturbation or on how the system couples to it. Similarly, Bis arbitrary and can be chosen at either scale. 3.5. Violation of the Fluctuation-Dissipation Theorem From the Volterra series expansion of the mean, Eq. (3.22), and of the covariance components, Eq. (3.23), we can derive a response relation that quantifies the violation of the fluctuation–dissipation theorem when the system is driven away from equilibrium by the multidimensional protocol. We begin by inserting the component-wise nonlinear fluctuation–dissipation relations, Eqs. (3.39)—(3.41), into β⟨Bt⟩and the time-integrated covariance β2Rt −∞ ds˙xi s⟨Bt;Fi s⟩. The time-integrated form of Eq. (3.23) is obtained by summing 27 3. Multidimensional Driving Protocols over the index that remains free in the original expression. This yields7 β⟨Bt⟩=β⟨B⟩eq +Zt −∞ ds˙xi sΓ(1,0) t−si+Zt −∞ dsZt −∞ ds′˙xi s˙xj s′Γ(1,1) t−s;t−s′ij −1 2Γ(2,0) t−s,t−s′ij+Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ Γ(1,2) t−s,t−s′;t−s′′ ijk +1 6Γ(3,0) t−s,t−s′,t−s′′ ijk −1 2Γ(2,1) t−s;t−s′,t−s′′ ijk+O((˙xi)4), (3.42) β2Zt −∞ ds˙xi s⟨Bt;Fi s⟩=Zt −∞ ds˙xi sΓ(0,1) t−si +Zt −∞ dsZt −∞ ds′˙xi s˙xj s′Γ(0,2) t−s,t−s′ij +1 2Γ(2,0) t−s,t−s′ij +Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ Γ(0,3) t−s,t−s′,t−s′′ ijk −1 6Γ(3,0) t−s,t−s′,t−s′′ ijk +1 2Γ(2,1) t−s;t−s′,t−s′′ ijk+O((˙xi)4). (3.43) By subtracting Eq. (3.43) from Eq. (3.42), and once again applying the componentwise nonlinear fluctuation–dissipation relations (3.39)—(3.41), we obtain β⟨Bt⟩ − β2Zt −∞ ds˙xi s⟨Bt;Fi s⟩= Γ(0,0) −1 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′Γ(2,0) t−s,t−s′ij +1 2Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ ×1 3Γ(3,0) t−s,t−s′,t−s′′ ijk −Γ(2,1) t−s;t−s′,t−s′′ ijk +O((˙xi)4). (3.44) Resorting to the Volterra series expansions, Eqs. (3.22)–(3.25), the right-hand side of Eq. (3.44) can be expressed entirely in terms of the components of the GCCFs, 7The index notation in the kernel components is adjusted accordingly due to summing over them. 28 3. Multidimensional Driving Protocols which are joint cumulants. This yields Γ(0,0) |{z} =β⟨B⟩eq −1 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′Γ(2,0) t−s,t−s′ij +Zt −∞ ds′′ ˙xk s′′ Γ(2,1) t−s;t−s′,t−s′′ ijk | {z } =β3⟨Bt;Fi s;Fj s′⟩ +1 6Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ Γ(3,0) t−s,t−s′,t−s′′ ijk | {z } =β4⟨Bt;Fi s;Fj s′;Fk s′′ ⟩ +O((˙xi)4) =β⟨B⟩eq −β3 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′⟨Bt;Fi s;Fj s′⟩ +β4 6Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ ⟨Bt;Fi s;Fj s′;Fk s′′ ⟩+O((˙xi)4) (3.45) Hence, the final relation, with both sides expressed entirely in terms of joint cumulants, reads: β⟨Bt⟩ − β2Zt −∞ ds˙xi s⟨Bt;Fi s⟩=β⟨B⟩eq −β3 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′⟨Bt;Fi s;Fj s′⟩ +β4 6Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ ⟨Bt;Fi s;Fj s′;Fk s′′ ⟩ +O((˙xi)4). (3.46) This response relation expresses the difference between the mean β⟨Bt⟩and the timeintegrated covariance β2Rt −∞ ds˙xi s⟨Bt;Fi s⟩in terms of time-integrated higher-order joint cumulants of Band the generalized force components. The summations incorporate the contributions from all ddriving components. The left-hand side resembles the FDT for a multidimensional perturbation and therefore vanishes identically8up to linear order in the driving velocity. The right-hand side consists of terms of higher order in the driving velocity components. Starting at second order, it shows that nonlinear and non-Gaussian fluctuations generate corrections to the linear multidimensional FDT. Hence, Eq. (3.46) can be understood as a fluctuation-identity, which quantifies the deviation from the FDT in such nonequilibrium situations. It is important to mention, that the right-hand side of Eq. (3.46) vanishes to arbitrary order in the driving velocity if all observables B, F1, . . . , Fdare Gaussian, since then all joint cumulants of order three or higher are zero [16,33]. In this scenario, the FDT remains valid even under strong driving. 8β⟨B⟩eq is part of the FDT. 29 4. Applications of the Formalism In this chapter, we apply parts of the derived formalism to two specific cases. First, we show that the one-dimensional driving case [16,24], which formed the foundation of this thesis, is encompassed by the generalization to multidimensional driving. Second, we consider a two-component velocity protocol and evaluate the corresponding response relation that quantifies the FDT violation. 4.1. One-Dimensional Driving The special case of a scalar protocol corresponds to d= 1 in our description. Hence, both the protocol vector and the generalized force vector contain only one component, which is why they can be treated as scalars. In particular, we have xs∈Rand Fs≡F(ys, xs)≡∂xsU(ys, xs)∈R. As a result, the GCCFs ⟨Bt;Ft1;. . . ;Ftm⟩are scalars as well, since Fis no longer a vector. Consequently, they are mathematically equivalent to joint cumulants. This means that for d= 1 the GCCFs reduce to the original CCFs and are no longer composed of components. Therefore, they can be directly expanded within the Volterra series framework. Accordingly, the memory kernels arising from these expansions need to be scalars as well. When we evaluate the Volterra series expansions, Eqs. (3.22)–(3.25) for d= 1 and the properties discussed above, we find1 β⟨Bt⟩= Γ(0,0) +Zt −∞ ds˙xsΓ(0,1) t−s+Zt −∞ dsZt −∞ ds′˙xs˙xs′Γ(0,2) t−s,t−s′ +Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xs˙xs′˙xs′′ Γ(0,3) t−s,t−s′,t−s′′ +. . . (4.1) β2⟨Bt;Ft′⟩= Γ(1,0) t−t′+Zt −∞ ds˙xsΓ(1,1) t−s;t−t′(4.2) +Zt −∞ dsZt −∞ ds′˙xs˙xs′Γ(1,2) t−s,t−s′;t−t′+. . . (4.3) 1The driving velocity ˙xs∈Ris now also a scalar quantity. 30 4. Applications of the Formalism β3⟨Bt;Ft′;Ft′′ ⟩= Γ(2,0) t−t′,t−t′′ +Zt −∞ ds˙xi sΓ(2,1) t−s;t−t′,t−t′′ +. . . (4.4) β4⟨Bt;Ft′;Ft′′ ;Ft′′′ ⟩= Γ(3,0) t−t′,t−t′′ ,t−t′′′ +. . . (4.5) . . . In accordance with the fact that all quantities have only one component, the expansions involve no summations. The explicit microscopic expressions for the appearing memory kernels can be obtained by evaluating Eqs. (3.27)–(3.30) and (3.32)–(3.37) for d= 1. It is important to note that in this case the symmetrization acts only on the time indices, since there are no component indices. Consequently, the component-wise nonlinear fluctuation–dissipation relations in Eqs. (3.39)–(3.41) also simplify. Since the memory kernels are scalars, the identities no longer connect components and involve only the kernels themselves. Furthermore, the symmetrization reduces to interchanges of the time arguments rather than the ordered pairs. They simplify to Γ(0,1) t= Γ(1,0) t(4.6) Γ(0,2) s1,s2=1 2X π∈S2 Γ(1,1) sπ(1);sπ(2) −1 2Γ(2,0) s1,s2(4.7) Γ(0,3) s1,s2,s3=1 6X π∈S3 Γ(1,2) sπ(1),sπ(2);sπ(3) −1 12 X π∈S3 Γ(2,1) sπ(1);sπ(2),sπ(3) +1 6Γ(3,0) s1,s2,s3.(4.8) . . . Finally, we evaluate the FDT violation as given by Eq. (3.46). We obtain β⟨Bt⟩ − β2Zt −∞ ds˙xs⟨Bt;Fs⟩=β⟨B⟩eq −β3 2Zt −∞ dsZt −∞ ds′˙xs˙xs′⟨Bt;Fs;Fs′⟩ +β4 6Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xs˙xs′˙xs′′ ⟨Bt;Fs;Fs′;Fs′′ ⟩ +O(˙x4), (4.9) as no summations remain in this case. These results for d= 1, obtained from the multidimensional driving formalism, are in exact agreement with those shown in Refs. [16,24]. Therefore, we conclude that the generalized formalism includes the already studied case of scalar driving. 31 4. Applications of the Formalism 4.2. Two-Component Velocity Protocol Next, we examine the special case of a two-component velocity protocol (d= 2), in which the system is driven simultaneously along the xand y-axis with constant velocities. Then the protocol velocity takes the form ˙ xs= ˙xx s ˙xy s ≡ vx vy with vx, vy=const. (4.10) Moreover, we consider a scenario in which the driving along the x-direction is significantly stronger than along the perpendicular y-direction, such that2vy≪vx. The FDT violation for this specific protocol can be investigated by evaluating Eq. (3.46). The left-hand side is given by β⟨Bt⟩ − β2Zt −∞ ds˙xi s⟨Bt;Fi s⟩=β⟨Bt⟩ − β2Zt −∞ dshvx⟨Bt;Fx s⟩+vy⟨Bt;Fy s⟩i, (4.11) where Fx sand Fy sdenote the generalized force components corresponding to the xand y-directions, respectively. Since the protocol velocity components are constant and thus time independent, they could in principle be moved outside the integrals when splitting the sum. However, for readability, we keep them inside the integrands. The terms of second and third order in the driving velocity, which make up the righthand side of Eq. (3.46), are given by −β3 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′⟨Bt;Fi s;Fj s′⟩ =−β3 2Zt −∞ dsZt −∞ ds′hv2 x⟨Bt;Fx s;Fx s′⟩+vxvy⟨Bt;Fx s;Fy s′⟩+⟨Bt;Fy s;Fx s′⟩ +v2 y⟨Bt;Fy s;Fy s′⟩i ≈ −β3 2Zt −∞ dsZt −∞ ds′hv2 x⟨Bt;Fx s;Fx s′⟩+vxvy⟨Bt;Fx s;Fy s′⟩+⟨Bt;Fy s;Fx s′⟩i, (4.12) 2We assume vx, vy= 0. 32 4. Applications of the Formalism β4 6Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ ⟨Bt;Fi s;Fj s′;Fk s′′ ⟩ =β4 6Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ hv3 x⟨Bt;Fx s;Fx s′;Fx s′′ ⟩+v2 xvy⟨Bt;Fx s;Fx s′;Fy s′′ ⟩ +⟨Bt;Fx s;Fy s′;Fx s′′ ⟩+⟨Bt;Fy s;Fx s′;Fx s′′ ⟩+vxv2 y⟨Bt;Fx s;Fy s′;Fy s′′ ⟩ +⟨Bt;Fy s;Fx s′;Fy s′′ ⟩+⟨Bt;Fy s;Fy s′;Fx s′′ ⟩+v3 y⟨Bt;Fy s;Fy s′;Fy s′′ ⟩i ≈β4 6Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ hv3 x⟨Bt;Fx s;Fx s′;Fx s′′ ⟩+v2 xvy⟨Bt;Fx s;Fx s′;Fy s′′ ⟩ +⟨Bt;Fx s;Fy s′;Fx s′′ ⟩+⟨Bt;Fy s;Fx s′;Fx s′′ ⟩i. (4.13) In both correction contributions we invoked the driving condition vy≪vx, neglecting terms quadratic in vyand of higher order. By making use of Eqs. (4.11)–(4.13) we can fully evaluate Eq. (3.46). We obtain β⟨Bt⟩ − β2Zt −∞ ds vx⟨Bt;Fx s⟩+β3 2Zt −∞ dsZt −∞ ds′v2 x⟨Bt;Fx s;Fx s′⟩ −β4 6Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ v3 x⟨Bt;Fx s;Fx s′;Fx s′′ ⟩ − β2Zt −∞ ds vy⟨Bt;Fy s⟩ =β⟨B⟩eq −β3 2Zt −∞ dsZt −∞ ds′vxvy⟨Bt;Fx s;Fy s′⟩+⟨Bt;Fy s;Fx s′⟩ +β4 6Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ v2 xvy⟨Bt;Fx s;Fx s′;Fy s′′ ⟩+⟨Bt;Fx s;Fy s′;Fx s′′ ⟩ +⟨Bt;Fy s;Fx s′;Fx s′′ ⟩+O(v4 x, v2 y), (4.14) which is the response relation quantifying the FDT violation for a system driven by this two-component constant velocity protocol. If the time-integrated covariance of Band Fxdoes not vanish, Rt −∞ ds vx⟨Bt;Fx s⟩ = 0, Eq. (4.14) can be rewritten in a more compact form. To this end, we introduce the new parameter β2 eff(vx)≡β2 1 + −β 2Rt −∞ dsRt −∞ ds′v2 x⟨Bt;Fx s;Fx s′⟩ Rt −∞ ds vx⟨Bt;Fx s⟩ + β2 6Rt −∞ dsRt −∞ ds′Rt −∞ ds′′ v3 x⟨Bt;Fx s;Fx s′;Fx s′′ ⟩ Rt −∞ ds vx⟨Bt;Fx s⟩ . (4.15) We refer to βeff as the effective inverse temperature. This parameter is a technically motivated definition, serving as a tool to quantify the system’s deviation from equilibrium and providing a measure of the strength of nonlinear and non-Gaussian 33 4. Applications of the Formalism fluctuations. It should not be regarded as a proper thermodynamic inverse temperature β= 1/(kBT), but merely as an indicator of how far the system is perturbed away from equilibrium [34,35]. Since the system is primarily driven along the xdirection, we define βeff to contain all x-only nonlinearities and therefore to depend exclusively on vx. Thus, βeff characterizes the deviation from equilibrium originating from the dominant driving component. We will see that this allows us to interpret the response relation in a new way. From Eq. (4.15) we observe that βeff =βis recovered for weak driving. In this case the system remains close to equilibrium, i.e. within the regime of linear response theory. This is consistent with our interpretation that βeff is a measure for the deviation from equilibrium. As already mentioned, the definition requires a non-vanishing linear response in the x-direction, Rt −∞ ds vx⟨Bt;Fx s⟩ = 0. If this condition is not satisfied and the leading contribution arises only at higher orders in the driving, this definition is no longer valid. In this case we must revert to the original response relation, Eq. (4.14). By making use of the newly defined βeff, Eq. (4.14) can be rewritten as β⟨Bt⟩ − β2 eff Zt −∞ ds vx⟨Bt;Fx s⟩ − β2Zt −∞ ds vy⟨Bt;Fy s⟩ =β⟨B⟩eq −β3 2Zt −∞ dsZt −∞ ds′vxvy⟨Bt;Fx s;Fy s′⟩+⟨Bt;Fy s;Fx s′⟩ +β4 6Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ v2 xvy⟨Bt;Fx s;Fx s′;Fy s′′ ⟩+⟨Bt;Fx s;Fy s′;Fx s′′ ⟩ +⟨Bt;Fy s;Fx s′;Fx s′′ ⟩+O(v2 y). (4.16) The structure of this response relation is interesting. By construction, we absorbed all nonlinear correction terms involving only the x-drive into the effective inverse temperature βeff . This restores an FDT-like structure for the x-response, where the linear response is now governed by βeff . Since the driving velocity in the y-direction is much smaller, all terms of order v2 y and higher can be neglected. Thus, the relevant y-response remains linear and is still controlled by the inverse bath temperature β. Therefore, the left-hand side of Eq. (4.16) can be interpreted as an effective FDT, where βeff governs the pure x-drive and βthe pure y-drive. From this point of view, the violation of this effective FDT is solely caused by the nonequilibrium coupling between the xand y-drives and is quantified by the right-hand side of Eq. (4.16) (beyond the equilibrium contribution β⟨B⟩eq). These two leading-order corrections 34 A. Joint Cumulants and Modified Response Relations As expected, the joint cumulant of a single random variable is just its mean, while the joint cumulant of two random variables corresponds to their covariance. Higherorder joint cumulants represent more complex correlation measures. A.2. Derivation of the Nonlinear Response via Equilibrium Joint Cumulants In the following, we briefly sketch how the modified response relation, Eq. (2.21), for the path observable is derived. We begin by decomposing the path response into its time-symmetric and time-antisymmetric components: ⟨O⟩=1 2⟨O+Oθ⟩ | {z } symmetric +1 2⟨O−Oθ⟩ | {z } antisymmetric .(A.6) By combining Eqs. (2.16) and (2.17), we obtain the two new response relations 1 2⟨O+Oθ⟩=⟨O⟩eq − ⟨D′O⟩eq −1 2D′′ −(D′)2−(S′)2 4Oeq −1 6D′′′ −3D′D′′ −3 4S′S′′ + (D′)3+3 4D′(S′)2Oeq +. . . , (A.7) 1 2⟨O−Oθ⟩=1 2⟨S′O⟩eq +1 2S′′ 2−D′S′Oeq +1 4S′′′ 3−D′S′′ −D′′S′+ (D′)2S′+(S′)3 12 Oeq +. . . (A.8) As before, these relations must hold for any path observable, and thus also for O= 1. This observable is trivially time-symmetric, and its average is independent of the system’s state, as it is always equal to 1. Therefore its nonequilibrium average is equal to its equilibrium one, ⟨1⟩=⟨1⟩eq. Applying O= 1 to Eqs. (A.7) and (A.8), while using the discussed properties, yields a set of identities: ⟨D′⟩eq =⟨S′⟩eq = 0,(A.9) D′′ −(D′)2−(S′)2 4eq = 0,(A.10) S′′ 2−D′S′eq = 0,(A.11) 41 A. Joint Cumulants and Modified Response Relations D′′′ −3D′D′′ −3 4S′S′′ + (D′)3+3 4D′(S′)2eq = 0,(A.12) S′′′ 3−D′S′′ −D′′S′+ (D′)2S′+(S′)3 12 eq = 0.(A.13) By using this list of identities, we can re-express the individual terms in Eq. (2.16) in terms of the joint cumulants from Eqs. (A.2)—(A.5). The first term can be expressed directly as −D′−S′ 2Oeq =1 2⟨S′O⟩eq − ⟨D′O⟩eq =1 2⟨S′O⟩eq +1 2⟨S′⟩eq | {z } =0 ⟨O⟩eq − ⟨D′O⟩eq − ⟨D′⟩eq | {z } =0 ⟨O⟩eq =1 2⟨S′;O⟩eq − ⟨D′;O⟩eq. (A.14) For the second term we find −1 2D′′ −S′′ 2Oeq −D′−S′ 22 Oeq=1 4⟨S′′O⟩eq −1 2⟨D′′O⟩eq +1 2⟨(D′)2O⟩eq −1 2⟨D′S′O⟩eq +1 8⟨(S′)2O⟩eq. (A.15) The terms on the right-hand side can be expressed in terms of joint cumulants. We find ⟨S′′O⟩eq =⟨S′′;O⟩eq +⟨S′′⟩eq⟨O⟩eq,(A.16) ⟨D′′O⟩eq =⟨D′′;O⟩eq +⟨D′′⟩eq⟨O⟩eq,(A.17) ⟨(D′)2O⟩eq =⟨D′;D′;O⟩eq +⟨(D′)2⟩eq⟨O⟩eq +⟨D′O⟩eq ⟨D′⟩eq | {z } =0 +⟨D′O⟩eq ⟨D′⟩eq | {z } =0 −2⟨D′⟩eq | {z } =0 ⟨D′⟩eq | {z } =0 ⟨O⟩eq,(A.18) ⟨D′S′O⟩eq =⟨D′;S′;O⟩eq +⟨D′S′⟩eq⟨O⟩eq +⟨D′O⟩eq ⟨S′⟩eq | {z } =0 +⟨S′O⟩eq ⟨D′⟩eq | {z } =0 −2⟨D′⟩eq | {z } =0 ⟨S′⟩eq | {z } =0 ⟨O⟩eq,(A.19) 42 A. Joint Cumulants and Modified Response Relations ⟨(S′)2O⟩eq =⟨S′;S′;O⟩eq +⟨(S′)2⟩eq⟨O⟩eq +⟨S′O⟩eq ⟨S′⟩eq | {z } =0 +⟨S′O⟩eq ⟨S′⟩eq | {z } =0 −2⟨S′⟩eq | {z } =0 ⟨S′⟩eq | {z } =0 ⟨O⟩eq.(A.20) Next, we sum up all of these terms and multiply them by the corresponding prefactors from Eq. (A.15). We obtain (already omitting all vanishing terms) 1 4⟨S′′O⟩eq −1 2⟨D′′O⟩eq +1 2⟨(D′)2O⟩eq −1 2⟨D′S′O⟩eq +1 8⟨(S′)2O⟩eq =1 4⟨S′′;O⟩eq −1 2⟨D′′;O⟩eq +1 2⟨D′;D′;O⟩eq −1 2⟨D′;S′;O⟩eq +1 8⟨S′;S′;O⟩eq −1 2D′′ −(D′)2−(S′)2 4eq | {z } =0 ⟨O⟩eq +1 2S′′ 2−D′S′eq | {z } =0 ⟨O⟩eq. (A.21) The final expression for the second term reads −1 2D′′ −S′′ 2Oeq −D′−S′ 22 Oeq=1 4⟨S′′;O⟩eq −1 2⟨D′′;O⟩eq +1 2⟨D′;D′;O⟩eq −1 2⟨D′;S′;O⟩eq +1 8⟨S′;S′;O⟩eq. (A.22) The third order term in Eq. (2.16) (and higher-order terms, which are already left out in the expansion) can be treated in the same way. Here, we restrict ourselves to the results presented, as the third-order term is already cumbersome to derive. Plugging Eqs. (A.14) and (A.22) into the path response, Eq. (2.16), yields ⟨O⟩=⟨O⟩eq +1 2⟨S′;O⟩eq − ⟨D′;O⟩eq +1 2h⟨D′;D′;O⟩eq − ⟨D′;S′;O⟩eqi −1 2⟨D′′;O⟩eq +1 4⟨S′′;O⟩eq +1 8⟨S′;S′;O⟩eq +. . . , (A.23) which is exactly Eq. (2.21). The nonlinear response of a state observable in terms of equilibrium joint cumulants, Eq. (2.22), can be derived in an analogous way from Eq. (2.20). The details are omitted here, as the derivation is essentially the same. 43 B. Derivation of the Volterra Series Expansions This appendix presents the derivations of the Volterra series expansions of the mean and covariance components. The expansions of the thirdand fourth-order GCCF components follow the same procedure as the covariance expansion and are therefore omitted for brevity. B.1. Expansion of the Mean The expansion of the mean β⟨Bt⟩can be formulated by evaluating Eq. (2.22) for the state observable Bt=B(yt,xt): ⟨Bt⟩=⟨Bt⟩eq +⟨S′;Bt⟩eq +1 2⟨S′′;Bt⟩eq − ⟨D′;S′;Bt⟩eq −1 2h⟨D′;S′′;Bt⟩eq +⟨D′′;S′;Bt⟩eqi+1 2⟨D′;D′;S′;Bt⟩eq +1 24⟨S′;S′;S′;Bt⟩eq +1 6⟨S′′′;Bt⟩eq +. . . (B.1) In the next step, we rewrite the individual terms by inserting the action components, Eqs. (3.13)–(3.17), and grouping them according to the driving order. From the resulting expressions, we can then identify the memory kernel components. The zeroth-order kernel is obtained directly as β⟨Bt⟩eq =β⟨B⟩eq = Γ(0,0),(B.2) where we used that equilibrium averages are time-independent. 44 B. Derivation of the Volterra Series Expansions The first order in the driving velocity is also given by a single term and reads β⟨S′;Bt⟩eq =Zt −∞ ds˙xi sβ2⟨Fi s;Bt⟩eq =Zt −∞ ds˙xi sβ2⟨Fi s−t;B0⟩eq =Zt −∞ ds˙xi sβ2⟨Fi t−s;B0⟩eq =Zt −∞ ds˙xi sΓ(0,1) t−si, (B.3) where the first-order memory kernel components can be identified as Γ(0,1) si=β2⟨Fi s;B0⟩eq.(B.4) In the calculation steps, we first used the determinacy of the protocol to take ˙xi soutside the covariance. Afterwards, we employed the stationarity of equilibrium timecorrelation functions [4] to introduce a time shift by −tand finally applied the timereversal symmetry of the equilibrium state, which was already used in Sec. 2.3.1. This requires taking into account how the observables in the joint cumulants transform under time reversal. The transformation properties follow from the action components, Eqs. (3.13)–(3.17), because S′, S′′, S′′′ need to be time-antisymmetric and D′, D′′ need to be time-symmetric. These operations will also be applied in all of the following calculations. The second-order contribution to the Volterra series expansion comes from the terms 1 2⟨S′′;Bt⟩eq − ⟨D′;S′;Bt⟩eq. For the first term, we find 1 2⟨S′′;Bt⟩eq =−Zt −∞ dsZt −∞ ds′˙xi s˙xj s′β∂Fi s ∂xj s ;Bteq Θ(s′−s).(B.5) On the other hand, we could also write 1 2⟨S′′;Bt⟩eq =−Zt −∞ dsZt −∞ ds′˙xi s˙xj s′β∂Fj s′ ∂xi s′ ;Bteq Θ(s−s′),(B.6) where we exchanged the ordered index pairs (s, i)↔(s′, j). Hence, it follows that β*∂Fi s ∂xj s ;Bt+eq Θ(s′−s) = β*∂Fj s′ ∂xi s′ ;Bt+eq Θ(s−s′),(B.7) 45 B. Derivation of the Volterra Series Expansions which shows that the kernel components must be symmetric in the ordered index pairs of (time, component). They are not symmetric under exchange of only the time indices or only the component indices, since swapping just one of them alters the velocity prefactors and thereby mismatches the kernel components, even though the total expression remains unchanged. By incorporating this symmetry property, we find β 2⟨S′′;Bt⟩eq −β⟨D′;S′;Bt⟩eq =−1 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′β2∂Fi s ∂xj s ;Bteq Θ(s′−s) −1 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′β2∂Fj s′ ∂xi s′ ;Bteq Θ(s−s′) −1 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′β2⟨Di s;Fj s′;Bt⟩eq −1 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′β2⟨Dj s′;Fi s;Bt⟩eq =Zt −∞ dsZt −∞ ds′˙xi s˙xj s′ β2 2⟨Di t−s;Fj t−s′;B0⟩eq +∂Fi t−s ∂xj t−s ;B0eq Θ(s′−s) +⟨Dj t−s′;Fi t−s;B0⟩eq +∂Fj t−s′ ∂xi t−s′ ;B0eq Θ(s−s′) =Zt −∞ dsZt −∞ ds′˙xi s˙xj s′ β2 2X π∈S2 ⟨Diπ(1) t−sπ(1) ;Fiπ(2) t−sπ(2) ;B0⟩eq +*∂Fiπ(1) t−sπ(1) ∂xiπ(2) t−sπ(1) ;B0+eq Θ(sπ(2) −sπ(1))  =Zt −∞ dsZt −∞ ds′˙xi s˙xj s′Γ(0,2) t−s,t−s′ij,(B.8) with the second-order memory kernel components Γ(0,2) s1,s2ij =β2 2X π∈S2 ⟨Diπ(1) sπ(1) ;Fiπ(2) sπ(2) ;B0⟩eq +*∂Fiπ(1) sπ(1) ∂xiπ(2) sπ(1) ;B0+eq Θ(sπ(1) −sπ(2)) .(B.9) The third-order contribution to the expansion arises from the remaining terms in Eq. (B.1). As for the second order, we symmetrize the expression with respect to the interchange of the ordered pairs. We obtain1 −β 2h⟨D′;S′′;Bt⟩eq +⟨D′′;S′;Bt⟩eqi+β 2⟨D′;D′;S′;Bt⟩eq +β 24⟨S′;S′;S′;Bt⟩eq 1Since the full expression is rather lengthy, we present the derivation here only in a compact form. 46 B. Derivation of the Volterra Series Expansions +β 6⟨S′′′;Bt⟩eq =Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ 1 6 β2*Di s;∂Fj s′ ∂xk s′ ;Bt+eq Θ(s′′ −s′) + . . . +β2*Dk s′′ ;∂Fj s′ ∂xi s′ ;Bt+eq Θ(s−s′)−β2 2⟨Di,j s,s′;Fk s′′ ;Bt⟩eq −. . . −β2 2⟨Dk,j s′′ ,s′;Fi s;Bt⟩eq +β2 2⟨Di s;Dj s′;Fk s′′ ;Bt⟩eq +· · · +β2 2⟨Dk s′′ ;Dj s′;Fi s;Bt⟩eq +β4 24⟨Fi s;Fj s′;Fk s′′ ;Bt⟩eq +· · · +β4 24⟨Fk s′′ ;Fj s′;Fi s;Bt⟩eq +β2 2*∂2Fi s ∂xj s∂xk s ;Bt+eq Θ(s′−s)Θ(s′′ −s) + . . . +β2 2*∂2Fk s′′ ∂xj s′′ ∂xi s′′ ;Bt+eq Θ(s′−s′′)Θ(s−s′′)  =Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ 1 6X π∈S3 β2 2 ⟨Diπ(1) t−sπ(1) ;Diπ(2) t−sπ(2) ;Fiπ(3) t−sπ(3) ;B0⟩eq +*∂2Fiπ(1) t−sπ(1) ∂xiπ(2) t−sπ(1) ∂xiπ(3) t−sπ(1) ;B0+eq Θ(sπ(2) −sπ(1))Θ(sπ(3) −sπ(1)) − ⟨Diπ(1),iπ(2) t−sπ(1),t−sπ(2) ;Fiπ(3) t−sπ(3) ;B0⟩eq +β4 24⟨Fiπ(1) t−sπ(1) ;Fiπ(2) t−sπ(2) ;Fiπ(3) t−sπ(3) ;B0⟩eq +β2*Diπ(1) t−sπ(1) ;∂Fiπ(2) t−sπ(2) ∂xiπ(3) t−sπ(2) ;B0+eq Θ(sπ(3) −sπ(2))  =Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ Γ(0,3) t−s,t−s′,t−s′′ ijk,(B.10) with the third-order memory kernel components Γ(0,3) s1,s2,s3ijk =1 6X π∈S3 β2 2 ⟨Diπ(1) sπ(1) ;Diπ(2) sπ(2) ;Fiπ(3) sπ(3) ;B0⟩eq +*∂2Fiπ(1) sπ(1) ∂xiπ(2) sπ(1) ∂xiπ(3) sπ(1) ;B0+eq Θ(sπ(1) −sπ(2))Θ(sπ(1) −sπ(3)) − ⟨Diπ(1),iπ(2) sπ(1),sπ(2) ;Fiπ(3) sπ(3) ;B0⟩eq +β4 24⟨Fiπ(1) sπ(1) ;Fiπ(2) sπ(2) ;Fiπ(3) sπ(3) ;B0⟩eq 47 B. Derivation of the Volterra Series Expansions +β2*Diπ(1) sπ(1) ;∂Fiπ(2) sπ(2) ∂xiπ(3) sπ(2) ;B0+eq Θ(sπ(2) −sπ(3)) .(B.11) By combining the results from Eqs. (B.2), (B.3), (B.8) and (B.10), we obtain the Volterra series expansion of the mean: β⟨Bt⟩= Γ(0,0) +Zt −∞ ds˙xi sΓ(0,1) t−si+Zt −∞ dsZt −∞ ds′˙xi s˙xj s′Γ(0,2) t−s,t−s′ij +Zt −∞ dsZt −∞ ds′Zt −∞ ds′′ ˙xi s˙xj s′˙xk s′′ Γ(0,3) t−s,t−s′,t−s′′ ijk +. . . (B.12) The expansion and the corresponding memory kernel components are exactly those presented in Sections 3.3.1 and 3.3.2. B.2. Expansion of the Covariance Components To derive the expansion of the covariance components β2⟨Bt;Fα t′⟩ =β2⟨B(yt,xt);Fα(yt′,xt′)⟩, we first need to expand Fα(yt′,xt′)around the reference equilibrium state, i.e., around xt′=xt. The expansion is obtained by slightly modifying Eq. (3.10): Fα(yt′,xt′) = Fα t′−Zt −∞ ds˙xi s ∂Fα t′ ∂xi t′ Θ(s−t′) +1 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′ ∂2Fα t′ ∂xi t′∂xj t′ Θ(s−t′)Θ(s′−t′) + . . . (B.13) At this point, we need to clarify the notation, which may seem a bit confusing but is necessary for presenting the kernel components in a compact form later on. On the left-hand side, we use the standard definition Fα t′=Fα(yt′,xt′). On the righthand side, however, we introduce the modified definition Fα t′≡Fα(yt′,xt′)|xt′=xt, as already done in Sec. 3.1. While this double use of the symbol Fα t′may at first seem problematic, only the standard definition on the left-hand side enters explicitly into the Volterra series expansions, whereas the modified definition on the right-hand side appears solely in the microscopic expressions for the memory kernels. In summary, the symbols must be read carefully, with attention to which side of the equation they appear on in the derivation. This notation was introduced in Refs. [14,16,24], and developing a clearer alternative may be a subtle but interesting point for future work. 48 B. Derivation of the Volterra Series Expansions Inserting Eq. (B.13) into the covariance components yields ⟨B(yt,xt);Fα(yt′,xt′)⟩=⟨Bt;Fα t′⟩ − Zt −∞ ds˙xi s*Bt;∂Fα t′ ∂xi t′+Θ(s−t′) +1 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′*Bt;∂2Fα t′ ∂xi t′∂xj t′+Θ(s−t′)Θ(s′−t′) +. . . (B.14) To proceed, we evaluate the three covariance terms that appear. The first takes the form ⟨Bt;Fα t′⟩=⟨BtFα t′⟩−⟨Bt⟩⟨Fα t′⟩.(B.15) The nonequilibrium averages can be further simplified by applying the nonlinear response relation for a path observable, Eq. (2.21). This yields ⟨BtFα t′⟩=⟨BtFα t′⟩eq −D′−S′ 2;BtFα t′eq +1 2h⟨D′;D′;BtFα t′⟩eq − ⟨D′;S′;BtFα t′⟩eqi −1 2⟨D′′;BtFα t′⟩eq +1 4⟨S′′;BtFα t′⟩eq +1 8⟨S′;S′;BtFα t′⟩eq +. . . , (B.16) ⟨Bt⟩⟨Fα t′⟩=⟨Bt⟩eq −D′−S′ 2;Bteq +1 2h⟨D′;D′;Bt⟩eq − ⟨D′;S′;Bt⟩eqi −1 2⟨D′′;Bt⟩eq +1 4⟨S′′;Bt⟩eq +1 8⟨S′;S′;Bt⟩eq +. . . ·⟨Fα t′⟩eq −D′−S′ 2;Fα t′+1 2h⟨D′;D′;Fα t′⟩eq − ⟨D′;S′;Fα t′⟩eqi−1 2⟨D′′;Fα t′⟩eq +1 4⟨S′′;Fα t′⟩eq +1 8⟨S′;S′;Fα t′⟩eq +. . . . (B.17) For the expansion of the covariance components, we restrict ourselves to terms up to second order in the driving velocity. In this case, inserting Eqs. (B.16) and (B.17) into Eq. (B.15) and re-expressing the result in terms of joint cumulants, we obtain ⟨Bt;Fα t′⟩=⟨Bt;Fα t′⟩eq −D′−S′ 2;Bt;Fα t′eq −1 2⟨D′′;Bt;Fα t′⟩eq +1 4⟨S′′;Bt;Fα t′⟩eq +1 2⟨D′;D′;Bt;Fα t′⟩eq +1 8⟨S′;S′;Bt;Fα t′⟩eq −1 2⟨D′;S′;Bt;Fα t′⟩eq. (B.18) The same method can be applied to the other two covariance terms in Eq. (B.14), 49 B. Derivation of the Volterra Series Expansions which then simplify to *Bt;∂Fα t′ ∂xi t′+=*Bt;∂Fα t′ ∂xi t′+eq −*D′−S′ 2;Bt;∂Fα t′ ∂xi t′+eq ,(B.19) *Bt;∂2Fα t′ ∂xi t′∂xj t′+=*Bt;∂2Fα t′ ∂xi t′∂xj t′+eq .(B.20) Consequently, the second and third term in Eq. (B.14) read −Zt −∞ ds˙xi s*Bt;∂Fα t′ ∂xi t′+Θ(s−t′) =−Zt −∞ ds˙xi s*Bt;∂Fα t′ ∂xi t′+eq Θ(s−t′) +Zt −∞ ds˙xi s*D′−S′ 2;Bt;∂Fα t′ ∂xi t′+eq Θ(s−t′), (B.21) 1 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′*Bt;∂2Fα t′ ∂xi t′∂xj t′+Θ(s−t′)Θ(s′−t′) =1 2Zt −∞ dsZt −∞ ds′˙xi s˙xj s′*Bt;∂2Fα t′ ∂xi t′∂xj t′+eq Θ(s−t′)Θ(s′−t′). (B.22) Combining Eqs. (B.18), (B.21) and (B.22), we obtain the final expansion. The calculation follows analogous steps as in the expansion of the mean: β2⟨Bt;Fα t′⟩=β2⟨Fα t−t′;B0⟩eq +Zt −∞ ds˙xi sβ2 Di t−s+βFi t−s 2;Fα t−t′;B0eq +*∂Fα t−t′ ∂xi t−t′ ;B0+eq Θ(s−t′)  +Zt −∞ dsZt −∞ ds′˙xi s˙xj s′ β2 2X π∈S2 1 2⟨Diπ(1) t−sπ(1) ;Diπ(2) t−sπ(2) ;Fα t−t′;B0⟩eq −1 2⟨Diπ(1),iπ(2) t−sπ(1),t−sπ(2) ;Fα t−t′;B0⟩eq +β 2⟨Diπ(1) t−sπ(1) ;Fiπ(2) t−sπ(2) ;Fα t−t′;B0⟩eq +β2 8⟨Fiπ(1) t−sπ(1) ;Fiπ(2) t−sπ(2) ;Fα t−t′;B0⟩eq + β 2*Fiπ(1) t−sπ(1) ;∂Fα t−t′ ∂xiπ(2) t−t′ ;B0+eq 50 Bibliography [12] J. C. P. Hollister, A. C. Wang, W. Kim, C.C Giza, M. L. Prins, and H. P. Kavehpour. Shear thinning behavior of cerebrospinal fluid with elevated protein or cellular concentration. Front. Phys., 11:1308136, 2015. [13] P. Oswald. Rheophysics: The Deformation and Flow of Matter. Cambridge University Press, 2009. [14] J. Caspers and M. Krüger. Nonlinear langevin functionals for a driven probe. J. Chem. Phys., 161(12):124109, 2024. [15] R. Kubo. The fluctuation-dissipation theorem. Rep. Prog. Phys., 29(1):255, 1966. [16] J. Caspers and M. Krüger. Identities for nonlinear memory kernels. Phys. Rev. E, 112(2):024124, 2025. [17] A. Ghadai and P. K. Bera S. Majumdar. Instability and stress fluctuations of a probe driven through a wormlike micellar fluid. J. Rheol., 69(5):599–609, 2025. [18] H. Seyforth, M. Gomez, W. B. Rogers, J. L. Ross, and W. W. Ahmed. Nonequilibrium fluctuations and nonlinear response of an active bath. Phys. Rev. Res., 4(2):023043, 2022. [19] R. Kubo. Statistical-mechanical theory of irreversible processes. i. general theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn., 12(6):570–586, 1957. [20] J. C. Dyre. Maximum-entropy ansatz for nonlinear-response theory. Phys. Rev. A, 40(4):2207–2210, 1989. [21] M. te Vrugt and R. Wittkowski. Mori-zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent hamiltonians. Phys. Rev. E, 99(6):062118, 2019. [22] U. Basu, M. Krüger, A. Lazarescu, and C. Maes. Frenetic aspects of second order response. Phys. Chem. Chem. Phys., 17(9):6653–6666, 2015. [23] M. Krüger and C. Maes. The modified langevin description for probes in a nonlinear medium. J. Phys. Condens. Matter, 29(6), 2016. 57 Bibliography [24] J. Caspers. Stochastic dynamics of driven Brownian particles in viscoelastic solvents. Phd thesis, Georg-August-Universität Göttingen, Göttingen, 2024. [25] J.-P. Hansen and I. R. McDonald. Theory of simple liquids: with applications to soft matter. Academic Press, 2013. [26] B. Müller. Brownian Particles in Nonequilibrium Solvents. Phd thesis, GeorgAugust-Universität Göttingen, Göttingen, 2019. [27] C. Maes. Response theory: a trajectory-based approach. Front. Phys, 8:229, 2020. [28] M. Schetzen. The Volterra and Wiener Theories of Nonlinear Systems. John Wiley & Sons Inc, 1980. [29] W. J. Rugh. Nonlinear System Theory: The Volterra-Wiener Approach. Johns Hopkins University Press, 1981. [30] C. Maes. Local detailed balance. SciPost Phys. Lect. Notes, 32, 2021. [31] A. Einstein. Die grundlage der allgemeinen relativitätstheorie. Annalen der Physik, 49(7):769–822, 1916. [32] D. E. Neuenschwander. Tensor Calculus for Physics: A Concise Guide. Johns Hopkins University Press, 2014. [33] G. Peccati and M. S. Taqqu. Wiener Chaos: Moments, Cumulants and Diagrams. Springer, 2011. [34] V. Démery and É. Fodor. Driven probe under harmonic confinement in a colloidal bath. J. Stat. Mech., 2019:033202, 2019. [35] L. F. Cugliandolo. The effective temperature. J. Phys. A: Math. Theor, 44(48): 483001, 2011. [36] J. Caspers, K. K. Kumar, C. Bechinger, and M. Krüger. Panoscopic nonequilibrium fluctuation identity. arXiv preprint, arXiv:2502.10175, 2025. 58 Acknowledgments First and foremost, I would like to express my gratitude to Prof. Dr. Matthias Krüger for giving me the opportunity to write this thesis in his group. His everopen door for questions and the guidance he provided have greatly deepened my understanding of nonequilibrium statistical physics and were essential in making this work possible. Riding together in the Tour d’Energie this year was a great experience. I would like to thank Juliana Caspers, whose work I had the privilege to extend a little further. She introduced me to the topic at the beginning, provided essential material to reproduce her results, and was always available via email. I wish her all the best in her new job in Bonn. Furthermore, I would like to thank Rupayan Saha, who looked after me during the past four months. Our many coffee chats, mostly about work but sometimes about the really important things in life, helped me a lot while writing this thesis. I truly appreciate that you always tried to answer my questions whenever I was stuck at a dead end. I would like to express my gratitude to my roommate Jacob. I will always remember the countless evenings we spent together on the balcony or in the kitchen, talking about silly things that made even the most stressful days a little less annoying. I am convinced that our flat will be much more organized in the future, but also a lot less fun. Special thanks go out to Nick, Max, Jannik, Ole, Thilo, Justin, Caroline, and all my other friends for making my time in Göttingen such a wonderful experience. Although it is sad to think that almost everyone is moving on, I will always remember the countless moments we laughed together and our many nights out. Lastly, I want to deeply thank my parents and my sister. Thank you for always being there for me in difficult times and for the love and support you give me every day. 59 Declaration on the use of ChatGPT and comparable tools In this thesis, I have used ChatGPT or comparable AI tools exclusively for proofreading, optimizing self-written text passages, and brainstorming. I hereby declare that I have stated all uses completely. 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 10. September 2025 (Julius Lüning)