Symmetry-induced fluctuation relations for dynamical observables irrespective of their behavior under time reversal

Symmetry-induced fluctuation relations for dynamical observables irrespective of their behavior under time-reversal Stefano Marcantoni,1, 2, ∗Carlos P´ erez-Espigares,3, 4, †and Juan P. Garrahan1, 2, ‡ 1School of Physics & Astronomy, University of Nottingham, Nottingham NG7 2RD, UK 2Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, UK 3Departamento de Electromagnetismo y F´ ısica de la Materia, Universidad de Granada, Granada 18071, Spain 4Institute Carlos I for Theoretical and Computational Physics, Universidad de Granada, Granada 18071, Spain We extend previous work to describe a class of fluctuation relations (FRs) that emerge as a consequence of symmetries at the level of stochastic trajectories in Markov chains. We prove that given such a symmetry, and for a suitable dynamical observable, it is always possible to obtain a FR under a biased dynamics corresponding to the so-called generalized Doob transform. The general transformations of the dynamics that we consider go beyond time-reversal or spatial isometries, and an implication is the existence of FRs for observables irrespective of their behaviour under time-reversal, for example for time-symmetric observables rather than currents. We further show how to deduce in the long-time limit these FRs from the symmetry properties of the generator of the dynamics. We illustrate our results with four examples that highlight the novel features of our work. I. INTRODUCTION Symmetries at the level of fluctuations or “fluctuation relations” (FRs) that hold far from equilibrium are one of the most general results of nonequilibrium statistical mechanics. First discovered at the end of the last century, with the celebrated Gallavotti-Cohen fluctuation theorem [1–3] being the prominent example, fluctuation relations represent the macroscopic footprint of a microscopic symmetry breaking by constraining the probability distribution of time-integrated observables for systems away from equilibrium. Since then, a lot of theoretical work has been devoted to the study of fluctuation relations both in the classical and in the quantum domain [4–19]. For reviews see [20–23]. Apart from the Gallavotti-Cohen fluctuation theorem - dealing with probabilities of an event and its time-reversal - other symmetries regarding spatial transformations, such as isometric fluctuation relations, have been unveiled in the last decade. This kind of relations were firstly introduced in the context of two-dimensional diffusive systems, by relating the probability of any pair of rotated currents [24] under some strong hypotheses that were subsequently removed [25]. The generalization to anisotropic systems [26] helped to test experimentally their validity by measuring the velocity fluctuations of a self-propelled rod [27] and those of hot Brownian swimmers [28]. This also triggered some works on the emergence of FRs for static observables in equilibrium systems with broken symmetries [29, 30]. Moreover, FRs for time-symmetric and activity-related quantities under involutions were discussed in [31, 32]. More recently, a thermodynamic uncertainty relation [33, 34] has been derived for fluxes that satisfy an isometric FR [35]. Although a spatial FR was introduced from a macroscopic perspective [24], its microscopic derivation was provided for Markovian stochastic systems in [36] under some ∗[email protected] †[email protected] ‡[email protected] assumptions on the dynamics. Here we build on the results of Refs. [36] and [31, 32] to generalise FRs that emerge as a consequence of symmetries in the dynamics. We do so in the framework of “thermodynamics of trajectories” [37–43], which extends the ensemble method of equilibrium statistical mechanics to dynamics. We show that given a dynamics which is symmetric under a certain transformation at the level of its trajectories, then a suitable observable can always be found that defines a related dynamics satisfying a FR. This new dynamics is one whose trajectory ensemble is exponentially biased with respect to the original one, which is achieved by means of a generalized Doob transform [44–48] that provides the optimal stochastic dynamics realizing a given fluctuation in the relevant observable. For long times, corresponding to the regime of large deviations [41], we show that from the symmetries of the generator it is possible to find the transformations which give rise to the FR. The paper is structured as follows. In Sect. II we review the basic formalism to study the statistics of trajectories in continuous-time Markov chains. By means of this formalism, we present in Sect. III the FR introduced in [36] discussing its hypotheses and showing how it can be generalized. In particular, we comment on the choice of the relevant observable and we point out that, given a symmetry of the original dynamics and a suitable observable, one can always obtain a FR by means of a proper conjugated dynamics through the generalized Doob transform. We further show how to obtain a FR from the symmetries of the generator. We also compare our findings with other results on time-symmetric observables already established in the literature. In Sect. IV we present four concrete examples which illustrate the novelty of our general results. Section V gives our conclusions. arXiv:2004.06173v2 [cond-mat.stat-mech] 2 Jul 2020 2 II. STATISTICS OF TRAJECTORIES AND GENERALIZED DOOB TRANSFORM For concreteness we focus on dynamics described by continuous-time Markov chains. Central to our analysis will be the framework known as “thermodynamics of trajectories” [37–43] whereby the standard ensemble method of equilibrium statistical mechanics is extended to ensembles of trajectories of the dynamics. A trajectory ωtup to time tis fully characterized by a sequence of configurations of the system x0, x1, x2, . . . , xNtogether with the times of jump between them t1, t2, . . . , tN: ωt:x0 t1 −→ x1 t2 −→ x2. . . xN−1 tN −→ xN. For simplicity we consider in the following systems with a finite number of configurations. The dynamics is determined by specifying the generator. This can be described using an operator formalism as, see e.g. [42, 49] ∂t|P(t)i=L|P(t)i with probability vector |P(t)i=PxP(x, t)|xi, where P(x, t) = hx|P(t)iis the probability of being in configuration xat time tand the generator Lreads L=X x,y6=x Wx→y|yihx|−X x Rx|xihx|,(1) with {|xi}being an orthonormal basis of configurations, such that hx|x0i=δx,x0. Here Wx→yare the jump rates between a pair of configurations xand y, and Rx=PyWx→yis the escape rate from configuration x. The probability of a certain trajectory ωtis then given by P(ωt)=e−(t−tN)RxNWxN−1→xN. . . e−t1Rx0Wx0→x1Px0, where Px0≡P(x0,0) is the probability of being in x0at t= 0. In this context, an observable is a functional on the trajectory space. It is customary to distinguish between two types of observables [40]: type-A observables are related to the jumps occurring in the trajectory, while type-B observables are related to the time spent in each configuration. More explicitly, type-A observables are of the form A(ωt) = X x,y Qx→y(ωt)αx→y,(2) where Qx→yis the number of jumps (or “flux”) from xto yin a trajectory ωtand αx→yare real parameters accounting for the contribution to the observable A(ωt)of each jump. For time-symmetric observables we have αx→y=αy→x. Instead, type-B observables are the time-integral of configurational functions, B(ωt) = Zt 0 dt0β[x(t0)] (3) with βbeing the quantity of interest evaluated in the configuration xat time t0. A typical example of type-A observable is the dynamical activity [39, 40, 42, 50], namely the total number of jumps in a trajectory. This corresponds to taking αx→y= 1 for any pair of connected configurations (x, y). An example of type-B observable is the time-integral of the magnetization in the trajectory of a spin system. The statistics of a stochastic observable K(ωt)can be retrieved by computing P(ωt)or alternatively from the moment generating function Z(s), that reads Z(s) = X ωt e−sK(ωt)P(ωt). The above equation is as well the normalization factor of the exponentially biased distribution Ps(ωt) = e−sK(ωt)P(ωt) Z(s),(4) known as the s-ensemble [51], which allows for the exploration of the rare events of interest through the parameter s. Using the operator formalism it can be shown that the moment generating function can be computed as the following scalar product [39–43], Z(s) = h−|etLs|P(0)i, where h−| =Pxhx|is the so-called flat state and the operator Lsis a tilted generator that reads Ls=X x,y6=x e−sαx→yWx→y|yihx|−X x Rx|xihx|,(5) for a type-A observable, or Ls=X x,y6=x Wx→y|yihx|−X xRx+sβ(x)|xihx|,(6) for a type-B observable. For large times the moment generating function satisfies a large deviation principle [39–43] Z(s)≈etθ(s), where θ(s)corresponds to the scaled cumulant generating function, which can be obtained as the largest eigenvalue of the tilted generator. At finite times the statistics of the selected observable depends on the full spectrum (and on the eigenvectors) of the tilted generator while at long times all the information concentrates in the largest eigenvalue. It is worth noting that the long-time average of the observable K(ωt)in the s-ensemble (4) is given by hK(ωt)is t=−θ0(s).(7) Unlike the original generator L, corresponding to the case s= 0, the tilted one is not a proper stochastic generator in the sense that it does not conserve probability, h−|Ls6= 0. However, it is possible to construct a proper stochastic generator (in general time-dependent) such that rare trajectories of the original process are mapped into typical trajectories of the new one. This is realized through the generalized Doob transform that produces the time-dependent generator LDoob t0(s) [46–48] LDoob t0(s) = Gt0LsG−1 t0−∂t0log(Z(s)) + (∂t0Gt0)G−1 t0 3 by means of the gauge transformation Gt0 Gt0=X x h−|e(t−t0)Ls|xi h−|e(t−t0)Ls|x0i|xihx|, where tis the final time. For asymptotically long times t−t01the exponential operator is well approximated as e(t−t0)Ls≃e(t−t0)θ(s)|r0ih`0|, where |r0iand h`0|are the right and left eigenvectors of Lscorresponding to the largest eigenvalue θ(s), namely h`0|Ls=θ(s)h`0|and Ls|r0i= θ(s)|r0i, which are normalized as h`0|r0i=h−|r0i= 1. Therefore, the gauge transformation becomes timeindependent and reads G∞=h`0|x0i−1Pxh`0|xi|xihx|so that in the end one has the following generator for long times [46–48] LDoob ∞(s) = G∞LsG−1 ∞−θ(s). Moreover, defining the matrix Lsas the matrix connecting the left eigenvector and the flat state h`0|=h−|Lswe get that Ls=G∞h`0|x0i. Thus assuming Lsis invertible, we can write the long-time Doob generator LDoob(s)≡ LDoob ∞(s)as LDoob(s) = LsLsL−1 s−θ(s),(8) which corresponds to a proper stochastic generator such that h−|LDoob(s)=0. One can show that the time-dependent Doob generator (8) describes an ensemble of stochastic trajectories with probability distribution given by (4), i.e. that is exponentially biased with respect to the ensemble generated by the original dynamics (see assumption 3 in the next section) [46, 47, 52]. The time-independent generator LDoob(s) generates instead Ps(ωt)in the long-time limit. In the examples of Sect. IV we will mainly use the time-independent Doob generator (8) as constructed above and comment on the time-dependent case in the second example. III. FLUCTUATION RELATION Fluctuation relations other than Gallavotti-Cohen such as FRs associated with spatial transformations were firstly introduced in the context of diffusive systems [24]. A derivation from the microscopic Markovian dynamics of this kind of FR was proved in [36] by means of three assumptions: 1. There is a bijection Rin the space of trajectories such that P0(Rωt) = P0(ωt), 2. There is an observable (maybe vectorial) K(ωt)such that K(Rωt) = U·K(ωt)for some matrix U(independent of ωt), 3. A modified dynamics exists such that the probability of a certain trajectory in this new dynamics is related to the probability under the original dynamics as follows PE(ωt) = e−ET·K(ωt)P0(ωt) Z0(E),(9) where Eis a field that breaks the initial symmetry and Z0(E) = Pωte−ET·K(ωt)P0(ωt)is the normalization. Here and in the following, column vectors are indicated by vand row vectors vT(with Tdenoting transposition), while the dot ·is the usual product of matrices. The FR is then expressed as a symmetry of the moment generating function ZE(λ)defined as follows ZE(λ) = X ωt e−λT·K(ωt)PE(ωt), describing the statistics of the stochastic observable Kin the modified dynamics. In particular, it turns out that ZE(λ) = ZE[(U−1)T·(λ+E)−E].(10) The proof of this result is quite straightforward. Indeed, by means of the three assumptions presented, one can write the following chain of equalities ZE(λ) = X ωt e−λT·K(ωt)PE(ωt) (3) =X ωt e−λT·K(ωt)e−ET·K(ωt)P0(ωt) Z0(E) (1) =X ωt e−(λ+E)T·K(ωt)P0(Rωt) Z0(E) =X ωt e−(λ+E)T·K(R−1ωt)P0(ωt) Z0(E) (3) =X ωt e−(λ+E)T·K(R−1ωt)eET·K(ωt)PE(ωt) (2) =X ωt e−(λ+E)T·U−1·K(ωt)+ET·K(ωt)PE(ωt) =X ωt e−(λ0)T·K(ωt)PE(ωt) = ZE(λ0),(11) where λ0= (U−1)T·(λ+E)−Eand the numbers parenthesis are used to clarify the role of each assumption. Remarkably, this relation is true at any finite time t. By looking at the behavior for asymptotically long times, one can also find out a symmetry relation at the level of the scaled cumulant generating function θ(λ). Indeed, since ZE(λ)∼etθE(λ)for long times, it turns out that θE(λ) = θE[(U−1)T·(λ+E)−E].(12) Our first contribution is to notice that the third assumption (9) is not an assumption, in the sense that, given a symmetry of P0(ωt)and a certain observable K(ωt), a dynamics satisfying Eq.(9) always exists. This dynamics is provided by the generalized Doob transform [44–48], briefly presented in the previous section, where the parameter s(that can be vectorial) has the role of the external field E. Thus, the Doob transform generates the ensemble of stochastic trajectories with probability PE(ωt). Therefore, the fluctuation relation given in (10) can be always found given the constant field E, which biases the statistics of trajectories P0(ωt)and which breaks its symmetry property. This relation includes as special cases 4 previously known results. For instance, if we choose the bijection Rto be the time-reversal and the observable K(ωt) to be a current (anti-symmetric under time reversal so that U amounts to a minus sign) we recover the celebrated GallavottiCohen relation for stochastic processes (see e.g. [11]). However, our result is more general inasmuch it deals also with transformations different from time-reversal, like spatial rotations and translations, and observables different from currents as for instance time-symmetric ones. In a couple of works [29, 30] a similar fluctuation relation was proved, by comparing the equilibrium Gibbs distributions relative to a symmetric Hamiltonian and a modified one where a field breaks the symmetry. Our work can be considered as a result along the same lines, where equilibrium ensembles in configuration space are replaced by dynamical ensembles in trajectory space. Also, our findings apply to equilibrium stochastic dynamics (when detailed balance is satisfied) as well as out-of-equilibrium. Active fluctuation symmetries are also discussed in the literature [32] pointing out that time-symmetric observables can also obey fluctuation relations [31]. In that context however, the analysis was limited to the study of involutions in the trajectory space. In this work instead we never use the assumption that the transformation Ris an involution. Following [32], we can use the previous framework also to derive another fluctuation relation for a generic observable f(ωt)in the modified dynamics hf(Rωt)iE=X ωt f(Rωt)PE(ωt) =X ωt f(Rωt)e−ET·K(ωt)P0(ωt) Z0(E) =X ωt f(ωt)e−ET·U−1K(ωt)P0(ωt) Z0(E) =X ωt f(ωt)e−(ET·U−1−ET)·K(ωt)PE(ωt) =hf(ωt) e−(ET·U−1−ET)·K(ωt)iE. Considering the constant function f(ωt) = 1 one obtains as a consequence a Jarzynski-like fluctuation relation De(ET−ET·U−1)·K(ωt)EE= 1,(13) and applying Jensen’s inequality this in turn gives a constraint on the average of the exponent D(ET−ET·U−1)·K(ωt)EE≤0,(14) or equivalently ET·DK(Rωt)EE≤ET·DK(ωt)EE.(15) This inequality provides a constraint on the average of the observable Kin the modified dynamics with respect to the same observable evaluated on the transformed trajectory. The results presented so far are valid for general bijections in the trajectory space. In the following, for the sake of convenience, we restrict the discussion to transformations at the configuration level. A. Choice of the observable We now consider the choice of the, in general vectorial, observable that satisfies the second assumption above. In particular, we show that it is always possible to find such an observable provided its dimension is sufficiently high and the transformation is actually a transformation in the configuration space. Consider a type-A observable A, as defined in (2), namely an observable related to the jumps between two configurations, whose components Aaare written as follows Aa(ωt) = X x,y Qx→y(ωt)αa x→y,(16) given the total number of jumps from xto yin the trajectory ωt,Qx→y(ωt), and a set of real parameters αa x→y. In a system with Dpossible configurations, in continuous time, the maximum number of allowed jumps is D(D−1), in the case of a fully connected problem. Therefore, a generic observable A belongs to a D(D−1) dimensional vector space, being a linear combination of the different number of jumps Qx→ywith real coefficients. Consider now a bijective transformation Racting on the configuration space. This in turn induces a map Rin the trajectory space given by ωt:x1→x2→. . . →xN R y Rωt:Rx1→Rx2→. . . →RxN.(17) As a consequence, the number of jumps between two configurations xand yin the original trajectory ωtequals the number of jumps between the transformed configurations Rx and Ry in the transformed trajectory Rωt, thus QRx→Ry(Rωt) = Qx→y(ωt).(18) The observable Ain the modified trajectory then reads Aa(Rωt) = X x,y Qx→y(Rωt)αa x→y =X x,y QR−1x→R−1y(ωt)αa x→y =X x,y Qx→y(ωt)αa Rx→Ry =e Aa(ωt),(19) where the first step is a consequence of (18) and the second one is just a change of variable. We want to find the linear transformation Uthat relates the original observable to e A(ωt) = U·A(ωt), with components e Aa(ωt) = X b UabAb(ωt) = X x,y Qx→y(ωt)X b Uabαb x→y. (20) 5 By comparing (19) and (20) one finds that U·αx→y=αRx→Ry ,(21) i.e. that the D2(D−1)2elements of the matrix Uhave to satisfy the system of linear equations PbUabαb x→y=αa Rx→Ry, for any pair x, y. These are in principle D2(D−1)2equations so that the system should allow for a solution. In order to better understand the condition for a unique solution we concentrate for the moment on the simplest case of a 2Dconfiguration space. Therefore, we consider a system with two possible configurations x, y so that just two different jumps (D(D−1) = 2) are possible x→yand y→x. Moreover one can have just two (D! = 2) different bijections R1and R2, where R1x=x, R1y=y, R2x=y, R2y=x. The maximun number of parameters αis D2(D−1)2= 4, indeed one has the four real parameters α1 x→y, α1 y→x, α2 x→y, α2 y→x, that allow us to write the equations for the matrix elements of Uas     α1 x→yα2 x→y0 0 α1 y→xα2 y→x0 0 0 0 α1 x→yα2 x→y 0 0 α1 y→xα2 y→x     | {z } M    u11 u12 u21 u22   =    α1 Rx→Ry α1 Ry→Rx α2 Rx→Ry α2 Ry→Rx    . Therefore, the solution is unique if and only if the matrix Mhas nonzero determinant, that in turn, due to the blockdiagonal structure of M, corresponds to have det(α)6= 0, where the matrix αis α=α1 x→yα2 x→y α1 y→xα2 y→x. The nonzero determinant implies the two components of the vectorial observable are indeed linearly independent. Otherwise one could recast them in a scalar observable and the dimensional argument would not work any more. The same reasoning holds true in higher dimensions so that it is always possible to construct a suitable observable (even though maybe not so relevant from a physical point of view) so that the assumption number 2is satisfied in a fully connected problem. If the system is not fully connected, one can restrict the previous discussion to the number of allowed jumps (strictly less than D(D−1)) and everything applies in the same way, thus implying the validity of assumption number 2. In this case, one has to be careful with the choice of the bijection Rin configuration space. Indeed, only those bijections Rthat preserve the set of allowed jumps induce a bijection Ron the trajectories of the system. This can be easily seen considering a totally asymmetric random walk on a ring (for simplicity let us just consider 4sites) 1→2 ↑ ↓ 4←3 A transformation Rsuch that R1=3, R2=2and R3=1 does not induce a bijection Rin the trajectory space of the system because for instance ωt: 1 →2→3should be mapped into eωt: 3 →2→1that is not allowed. Instead, a transformation like Rx =x+ 1(MOD 4), preserves the set of allowed jumps and is therefore acceptable. This will be the situation discussed in the examples of Section IV. More precisely, in those examples we will show that it is usually possible and more interesting to find low dimensional observables satisfying the assumption number 2. Further comments on the construction of low dimensional observables can be found in Appendix A. B. FR from the symmetries of the generator The FR (10) presented above is very compelling as it relates the probability of different fluctuations at all times from a symmetry of the probability of trajectories, P0(ωt) = P0(Rωt). However, for bijective transformation acting on configurations—as in (17)—the symmetry P0(ωt) = P0(Rωt)holds when both the transition rates and the probability of the initial state are symmetric under the transformation, namely Wx→y=WRx→Ry and Px0=PRx0. This might be something difficult to have, since we should prepare the system in an initial symmetric state. Yet, we show in this section that we can derive a FR for long times, i.e. θE(λ) = θE[(U−1)T·(λ+E)−λ], just from the symmetries of the dynamical generator (1) –so that Wx→y=WRx→Ry–, without caring about the symmetries of the initial state. We thus start by assuming that the original generator (1) has a certain symmetry under the transformation R, described by the operator Vsuch that V|xi=|Rxi, L=VLVT,(22) that in turn implies Wx→y=WRx→Ry for any pair (x, y). From this we now prove the following similarity transformation for the tilted generator Lλ=VL(U−1)T·λVT.(23) Since the tilted generator with respect to a type-A observable is Lλ=X x,y6=x e−λT·αx→yWx→y|yihx|−X x Rx|xihx|, the transformed one thus reads VLλVT=X x,y6=x e−λT·αx→yWx→y|RyihRx|−X x Rx|RxihRx|. By applying a change of variable and using the fact, as shown in (21), that the parameters αx→ytransform according to αR−1x→R−1y=U−1·αx→y the relation (23) follows immediately. Then, denoting Ls for the diagonal matrix whose entries corresponds to the left 6 eigenvector associated with θ(s), and exploiting the relation between the tilted generator and the Doob one for long times [46, 53] LDoob λ(s) = LsLλ+sL−1 s−θ(s),(24) we also find a symmetry relation at the level of the tilted Doob generator (24): such generator describes the statistics of the observable in the Doob dynamics and satisfies the following symmetry relation, LDoob λ(s) = AsLDoob (U−1)T·(λ+s)−s(s)A−1 s(25) where As=LsV L−1 s. Identifying swith the field Ewe see that (25) implies the FR (12), θE(λ) = θE[(U−1)T·(λ+E)−λ].(26) The same result can be derived for type-B observables (3), for which the tilted generator is given by (6). We thus have demonstrated that from the symmetries of the generator for bijective transformations Racting on configurations such that K(Rωt) = U·K(ωt), the FR (26) is derived. IV. EXAMPLES We now consider four different examples of increasing complexity to illustrate the general FRs obtained above. The first two examples are analytically solvable and deal with a single particle hopping on a ring. The first one shows that a fluctuation relation can exist for an activity-like observable (symmetric under time-reversal) provided it can take both positive and negative values. The second example deals with a two-dimensional (time-symmetric) observable that only has positive entries. Indeed, in this case, the increased dimensionality is sufficient to provide the symmetry of the scaled cumulant generating function. The other two examples involve many-body dynamics and are related to the fluctuation of the magnetization, a type-B observable. In particular, we show that a fluctuation relation holds true for the time-integrated magnetization in a Glauber-Ising dynamics modified with a transverse field. By looking at the generator, we also discuss the same observable in the context of a three-state Potts model. A. Time-symmetric observable for an asymmetric random walk Consider a particle performing an asymmetric random walk on a ring of Lsites. The particle jumps to the right with rate γ1and to the left with rate γ2. The net number of jumps in a given trajectory corresponds to the time-integrated current, while the total number of jumps is the activity. However, in order to illustrate the FR derived above we focus on a timeextensive observable that is time-symmetric, but with the possibility to take positive and negative values. Therefore the observable we choose is K(ωt) = Keven(ωt)−Kodd(ωt), namely the difference between the number of jumps in even and odd bonds in a given trajectory. The mean stationary value of this observable is zero, since there is no asymmetry between bonds as the hopping rates are the same for any bond. We choose Leven for convenience so that we have the same number of even and odd bonds. The exponentially tilted generator of the process thus reads Lλ=γ1e−λX xeven |x+ 1ihx|+γ2e−λX xodd |x−1ihx|+ +γ1eλX xodd |x+ 1ihx|+γ2eλX xeven |x−1ihx| − (γ1+γ2) 1 where |xiis the configuration in which the particle is at site x∈ {1,2, ..., L}. This describes a situation where four kinds of jump are weighted differently, namely, apart from distinguishing clockwise and counterclockwise jumps as in the original process, the rate depends on the kind of bond being even or odd. From Lλwe see that positive values of λbias the dynamics towards a negative value of K, by enhancing the number of jumps in odd bonds, while negative values of λdo the opposite (see Fig. 1, which is explained below). This generator can be diagonalized exactly. The eigenvalues ξqsatisfy the relation (ξq+γ1+γ2)2= 2γ1γ2cosh(2λ) + γ2 1e−2iq +γ2 2e2iq, where q=2πn Lwith n∈ {0,1, . . . , L −1}. The right eigenvectors can be written as |rq(λ)i= L/2 X m=1 hei2mqc2m|2mi+ ei(2m−1)qcq 2m−1|2m−1ii, where c2m=cis a constant ∀mand the odd coefficients cq 2m−1read cq 2m−1=cγ1e−λe−iq +γ2eλeiq (2γ1γ2cosh(2λ) + γ2 2e2iq +γ2 1e−2iq)1/2.(27) The left eigenvectors are obtained by exchanging qwith −q and γ1with γ2. As a result of the normalization conditions hlq|rq0i=δqq0and h−|r0i= 1 one finds that c= 1/L. As a first check we can see that the scaled cumulant generating function θ(λ) = −(γ1+γ2) + qγ2 1+γ2 2+ 2γ1γ2cosh(2λ)(28) is vanishing for λ= 0 and satisfies the symmetry property θ(λ) = θ(−λ). This symmetry is displayed in Fig. 1, where we show θ(λ)together hKiλ/t for γ1=γ2= 1. This is indeed the expected behaviour due to the properties of the original generator L0that is symmetric under the shift of one site L0=VL0VT,V|xi=|x+1i, and due to the chosen observable that instead changes sign under the same transformation (U=−1). The same is true for a shift of any odd number of sites. We can now perform the Doob transform in order to find a proper stochastic generator where the initial symmetry is 7 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12  <latexit sha1_base64="ucBmG3sPfYl2TYZJjrXfarKcHak=">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</latexit> >0 <latexit sha1_base64="lrS1K0t1Mkjq+gtD+FP947WWE8w=">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</latexit> <0 <latexit sha1_base64="LdMb/rE0gy8sgldH2YOAVnphxiI=">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</latexit> =0 <latexit sha1_base64="pAMDjH9HwBCGQCvRRzZAqpbNvjc=">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</latexit> ✓() <latexit sha1_base64="A3LXmyddXCaBH207eg8Q0dmHEpo=">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</latexit> hKi t <latexit sha1_base64="T8KMMGy3Jw40pDRoyu7Ab711MFE=">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</latexit> 00.5 1 -0.5-1 0 -1 -2 2 1 FIG. 1. Random walk on a ring. Scaled cumulant generating function (28) (solid blue line) with γ1=γ2= 1 for the observable K, namely the difference between the number of jumps in even and odd bonds, together with hKiλ/t =−θ0(λ)(red dashed line). Insets: Sketch of the Doob dynamics of the system (29) for s=λ and different values of λ. Thicker arrows correspond to larger transition rates (e|λ|) while thinner arrows correspond to smaller transition rates (e−|λ|). explicitly broken LDoob(s) = γ1e−sα(s)X xeven |x+ 1ihx|+ +γ2e−sα(s)−1X xodd |x−1ihx|+γ1esα(s)−1X xodd |x+ 1ihx|+ +γ2esα(s)X xeven |x−1ihx| − qγ2 1+γ2 2+ 2γ1γ2cosh(2s) 1 , (29) with α(s)being the following ratio α(s) = γ1es+γ2e−s pγ2 1+γ2 2+ 2γ1γ2cosh(2s).(30) Notice that for γ1=γ2=γwe get α(s)=1, so that the rates in the biased stochastic dynamics given by (29) become γe−sfor clockand counter-clockwise jumps over even bonds and γes, also in both directions, for odd bonds. This has been sketched in the insets to Fig. 1 for γ= 1 and s=λ, where the parity of the bonds has been made explicit. By exponentially tilting the previous generator one can uncover the fluctuation relation in the modified dynamics. Indeed, one has explicitly LDoob λ(s) = γ1e−(s+λ)α(s)X xeven |x+ 1ihx|+ +γ2e−(s+λ)α(s)−1X xodd |x−1ihx|+ +γ1es+λα(s)−1X xodd |x+ 1ihx|+ +γ2es+λα(s)X xeven |x−1ihx|+ −qγ2 1+γ2 2+ 2γ1γ2cosh(2s) 1 , so that at the level of the generator it turns out that LDoob λ(s) = AsLDoob −λ−2s(s)A−1 s,(31) where the transformation As(·)A−1 sexchanges the terms α(s)Pxeven |x+1ihx|and α(s)−1Pxodd |x+1ihx|and preserves the spectrum. Therefore, the symmetry on the scaled cumulant generating function reads θDoob(λ) = θDoob(−λ−2s).(32) This can indeed be easily verified from the explicit expression of θDoob(λ) θDoob(λ) = qγ2 1+γ2 2+ 2γ1γ2cosh(2s+ 2λ) −qγ2 1+γ2 2+ 2γ1γ2cosh(2s).(33) B. Two-dimensional observable for a totally asymmetric random walk Consider a totally asymmetric random walk, namely a particle hopping clockwise with rate γon a ring of Lsites. As in the previous example, the configuration is completely specified at any time by the position of the particle in the lattice. Let us consider now a vectorial observable K(ωt) = Keven(ωt) Kodd(ωt),(34) where Kodd (Keven) is the number of jumps from odd (even) sites, and a transformation Racting on the configurations that translates the position in the lattice by an odd number of sites. This transformation at the trajectory level induces a transformation of the observable described by a matrix U. Explicitly, since we are exchanging even and odd sites, the map can be written as follows U=0 1 1 0, K(Rωt) = U·K(ωt).(35) The exponentially tilted (relatively to the observable K) generator of the stochastic process reads Lλ=γe−λ1X xeven |x+ 1ihx|+γe−λ2X xodd |x+ 1ihx|−γ 1 . (36) In this case, negative λ1(λ2) enhance jumps starting from even (odd) sites. and positive values of the biasing field do the opposite. The tilted generator can be diagonalized exactly. Indeed, by assuming the following ansatz for the right eigenvector |rq(λ)icorresponding to the eigenvalue ξq(λ) |rq(λ)i= L−1 X m=0 cm(λ)eimq|mi,(37) one arrives at a system of coupled linear equations for the coefficients cm(λ). In particular, for m∈ {1, . . . , L/2}one has 8 two different sets of equations corresponding to even and odd jumps (c2m(λ)ξq(λ) + γ−γe−λ2e−iqc2m−1(λ) = 0, c2m−1(λ)ξq(λ) + γ−γe−λ1e−iqc2m−2(λ) = 0, (38) that in turn result into c2m(λ)ξq(λ) + γ2−c2m−2(λ)γ2e−(λ1+λ2)e−i2q= 0. (39) By summing over mand exploiting the periodic boundary conditions, one arrives at L/2 X m=1 c2m(λ)hξq(λ) + γ2−γ2e−(λ1+λ2)e−i2qi= 0.(40) Assuming PL/2 m=1 c2m(λ)6= 0 (we have checked this for consistency a posteriori) it turns out that the eigenvalues are ξq(λ) = γe−λ1+λ2 2−iq −1,(41) with q=2πn Land ntaking values in {0,1, . . . , L−1}. Therefore one can access the scaled cumulant generating function that is θ(λ) = γe−λ1+λ2 2−1(42) and check that indeed it satisfies the fluctuation relation θ(λ) = θ[(U−1)T·λ],(43) since (U−1)T=Uand it just consists in exchanging λ1 and λ2. Actually, all the points in the λ1, λ2plane such that λ1+λ2=c, with constant c, have the same value of θ(λ). We show this in Fig. 2, where some of the isolines have been displayed. By means of equations (38) one can explicitly compute the right eigenvectors of the tilted generator. In particular one finds that c2m−1=rand c2m=reλ1−λ2 2for some normalization constant r. Correspondingly, the left eigenvectors are represented as follows h`q(λ)|= L−1 X n=0 e−inq˜cn(λ)hn|,(44) where the coefficients read ˜c2m−1=`and ˜c2m=`eλ2−λ1 2 with normalization constant `. The orthonormality condition h`q0|rqi=δqq0fixes the product r ` to be 1/L. The further condition h−|r0(λ)i= 1 can be used to fix rand `separately. In particular, one finds that r=2 L 1 1+eλ2−λ1 2 .(45) We can compute the moment generating function Z(λ)by means of the relation Z(λ) = h−|etLλ|x0iand it reads Z(λ) = (e−γt cosh (η)+eλ1−λ2 2e−γt sinh (η),even x0, e−γt cosh (η)+eλ2−λ1 2e−γt sinh (η),odd x0, 0 0.5 1-1 -0.5 0 -1 -0.5 0.5 1 -1 0 1 2 1 <latexit sha1_base64="k8vC/mNf5TX9DLlPe7AQ7barQws=">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</latexit> 2 <latexit sha1_base64="U6erGdErcwtDNecSE/N9NUlkBY0=">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</latexit> -1 -0.5 0 0.5 1 1.5 2 FIG. 2. Totally asymmetric random walk. Scaled cumulant generating function θ(λ)given by (42). The symmetry θ(λ) = θ(U·λ) can be observed by looking at some points (λ1, λ2) and their symmetric pairs (given by symbols) with respect to λ1=λ2(black dashed line). We further notice that θ(λ)is invariant for all λ1+λ2= constant: e.g. λ1+λ2=−1,0,1(white solid lines). where the variable ηhas been defined as η=γt e−λ1+λ2 2.(46) One can find the time-dependent Doob transform that is related to the following gauge transformation [48] Gt0=X x h−|e(t−t0)Lλ|xi h−|e(t−t0)Lλ|x0i|xihx|= =(PL/2 m=1 (|2m−1ih2m−1|+g(t, t0, λ)|2mih2m|), PL/2 m=1 g−1(t, t0, λ)|2m−1ih2m−1|+|2mih2m|, where the first line refers to odd x0and the second one to even x0and the function g(t, t0, λ)explicitly reads g(t, t0, λ) = 1+es2−s1 2tanh hγ(t−t0)e−s1+s2 2i 1+es1−s2 2tanh hγ(t−t0)e−s1+s2 2i.(47) Given this transformation, the evolution of an odd initial configuration is given by the time-ordered exponential of the following time-dependent stochastic generator LDoob t0(s) = L/2−1 X m=0 γe−s1g−1|2m+ 1ih2m|+ +γe−s2g|2m+ 2ih2m+ 1|+∂t0g g|2mih2m|+ −(γ+∂t0log Zt0) 1 ,(48) where the dependence on t, t0, s in the function ghas been omitted to ease the notation. This generator describes a process where one has a different transition rate for even and odd 9 jumps, so that the symmetry of the original dynamics is broken. Unfortunately, due to time ordering, the calculation of the evolution of the generic configuration |x0iis too complicated. However, by looking at the explicit expressions of the functions g(t, t0, s)one can show that for long enough times, tt01, the Doob generator tends to a time-independent generator of the form LDoob(s) = γe−(s1+s2)/2X x|x+ 1ihx|− 1 ,(49) namely it describes just a rescaling of the original process and the symmetry is restored. This can be physically understood since the number of odd and even jumps in the long time limit tend to be equal irrespectively of the different jumps rates. One could have obtained the same result by directly computing the usual (time-independent) Doob transform. Indeed, by looking at the tilted generator (36) and using the largest eigenvalue θ(λ)(42) and the corresponding left eigenvector that has components `2n−1= 1 and `2n= e(λ1−λ2)/2one gets LDoob(s) = γe−s1X xeven `x+1 `x (s)|x+ 1ihx|+ +γe−s2X xodd `x+1 `x (s)|x+ 1ihx|−γ+θ(s) 1 =γe−(s1+s2)/2X x|x+ 1ihx|− 1 .(50) By means of an exponential tilting one arrives at the following scaled cumulant generating function θDoob λ(s) = −γe−(s1+s2)/21−e−(λ1+λ2)/2.(51) Given the parameter λ0defined as λ0= (U−1)T·(λ+s)−s=λ2+s2−s1 λ1+s1−s2,(52) one easily verifies the fluctuation relation θDoob(λ) = θDoob(λ0). The Jarzynski-like relation (13) in this case reads he(s1−s2)(Keven−Kodd)i= 1,(53) which means (s1−s2)hKeven−Koddi ≤ 0. The interpretation of this result is quite straightforward: if s1> s2the field suppresses the probability of even jumps with respect to odd jumps. This is confirmed by looking at the ratio between the even jump rate e−s1g−1and the odd jump rate e−s2gthat is always smaller than one, for s1> s2, and tends to one for long times. Indeed, one has e−s1g−1 e−s2g= es2−s1g−2= = es2−s1 2+ tanh hγ(t−t0)e−s1+s2 2i 1+es2−s1 2tanh hγ(t−t0)e−s1+s2 2i  2 ≤1,(54) because a+b≤1 + ab for any 0≤a≤1and 0≤b≤1. C. One-dimensional Ising model We now derive the fluctuation relation for a type-B observable, such as the time-integrated magnetization M(ωt) = Zt 0 m(t0)dt0,with m(t) = L X k=1 sk(t)(55) of a one-dimensional Ising model of Lsites with periodic boundary conditions and undergoing Glauber dynamics [54]. As a transformation Von the generator, we consider flipping all spins. The Hamiltonian of the system is H= −JPL k=1 sksk+1, with Jbeing the interaction constant and spin values sk∈ {−1,1}. Every configuration of the system {C}={sk}k=1,...,L is represented as a vector in a Hilbert space, |Ci= L O k=1 1+sk 2 1−sk 2,(56) such that sk= 1 corresponds to |1ik= (1,0)Tand sk= −1to |0ik= (0,1)T. Thus the probability of the system at time tis encoded in the vector |P(t)i=P2L i=1 P(Ci, t)|Cii, with P(Ci, t)standing for the probabilities of the different configurations Ciat time t. The evolution equation for the system with the Glauber dynamics is thus given by ∂t|P(t)i=L|P(t)i, where [49, 54] L=1 2 L X k=1 (σx k− 1 )h 1 −γ 2σz k(σz k−1+σz k+1)i,(57) with γ= tanh(2βJ)and inverse temperature β= 1/(kBT). Here σx,z kare the standard Pauli matrices acting on site k, and 1 is the 2L×2Lidentity matrix 1 =NL k=1 1 k. The Glauber generator (57) encodes the spin flip at site kby means of the σx koperator at a rate given by 1/2−γsk(sk−1+sk+1)/4. In order to bias the original generator to have a given timeintegrated magnetization, we firstly write the magnetization in an operatorial form as ˆm= L X k=1 σz k,(58) so that the tilted generator is then Lλ= L X k=1 1 2(σx k− 1 )h 1 −γ 2σz k(σz k−1+σz k+1)i−λσz k. (59) Notice that for negative (positive) λwe are biasing the system towards a positive (negative) magnetization. At this point it is easy to check that the original generator (57) is symmetric under the spin flipping transformation, namely if we take V= L Y k=1 σx k