Fluctuation, Dissipation and the Arrow of Time

The recent development of the theory of fluctuation relations has led to new insights into the ever-lasting question of how irreversible behavior emerges from time-reversal symmetric microscopic dynamics. We provide an introduction to fluctuation relations, examine their relation to dissipation and discuss their impact on the arrow of time question.

. Autonomous vs. nonautonomous dynamics. Top: Autonomous evolution of a gas from a non-equilibrium state to an equilibrium state (Minus-First Law). Bottom: Nonautonomous evolution of a thermally isolated gas between two equilibrium states. The piston moves according to a pre-determined protocol specifying its position λ t in time. The entropy change is non-negative (Second Law).
The Minus-First Law of Thermodynamics and the Second Law of Thermodynamics consider two very 21 different situations, see Fig. 1. The Minus-First Law deals with a completely isolated system that begins 22 in non-equilibrium and ends in equilibrium, following its spontaneous and autonomous evolution. In the 23 Second Law one considers a thermally (but not mechanically) isolated system that begins in equilibrium.

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A time-dependent mechanical action perturbs the initial equilibrium, the action is then turned off and a 25 final equilibrium will be reached, corresponding to higher entropy. 1 At variance with the Minus-First 26 Law, here the system does not evolve autonomously, but rather in response to a driving: we speak in this 27 case of nonautonomous evolution. 28 The use of the qualifiers "autonomous" and "nonautonomous" reflects here the fact that the set of 29 differential equations describing the microscopic evolution of the system are autonomous (i.e. they do 30 not contain time explicitly) in cases of the type depicted in Fig. 1, top, and are nonautonomous (i.e. 31 they contain time explictely) in cases of the type depicted in Fig. 1, bottom. Accordingly the Hamilton 32 function is time independent in the former cases and time dependent in the latter ones (see Sec. 2 below). 33 In order to illustrate the necessity of clearly distinguishing between the two prototypical evolutions 34 depicted in Fig. 1, let us analyze one statement which is often referred to as the second law: after the 35 1 That such final equilibrium state exists is dictated by the Minus-First Law. Here we see clearly the reason for assigning a higher rank to the Equilibrium Principle removal of a constraint, a system that is initially in equilibrium reaches a new equilibrium at higher 36 entropy [4]. While, after the removal of the constraint the system evolves autonomously (hence, in 37 accordance to the equilibrium principle will eventually reach a unique equilibrium state), it is often over-38 looked the fact that the overall process is nevertheless described by a set of nonautonomous differential 39 equations (because the removal of the constraint is an instance of a external time-dependent mechanical 40 intervention) with the constrained equilibrium as initial state. Then, in accordance with Clausius princi-41 ple the final state is of higher or same entropy. Thus, this formulation of the second law can be seen as  The field of fluctuation theorems has recently gained much attention. Many fluctuation theorems have 56 been reported in the literature, referring to different scenarios. Fluctuation theorems exist for classical 57 dynamics, stochastic dynamics, and for quantum dynamics; for transiently driven systems, as well as 58 for non equilibrium steady states; for systems prepared in canonical, micro-canonical, grand-canonical 59 ensembles, and even for systems initially in contact with "finite heat baths" [8]; they can refer to dif-60 ferent quantities like work (different kinds), entropy production, exchanged heat, exchanged charge, 61 and even information, depending on different set-ups. All these developments including discussions of Here (q, p) = (q 1 . . . q f , p 1 . . . p f ) denotes the conjugate pairs of coordinates and momenta describing 75 the microscopic state of the system.

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The assumption of time-reversal symmetry implies that if [q(t), p(t)] is a solution of Hamilton equa-77 tions of motion, then, for any τ , is also a solution of Hamilton equations of motion.

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This is the well known principle of microreversibility for autonomous systems [12]. 79 We assume that the system is at equilibrium described by the Gibbs ensemble: where Z(β) = dpdqe −βH(q,p) is the canonical partition function, and β −1 = k B T , with k B being the 81 Boltzmann constant and T denotes the temperature. 82 We next imagine to be able to observe the time evolution of all coordinates and momenta within some Γ is observed. We will reserve the symbol Γ to denote the whole trajectory (that is, mathematically 85 speaking, to denote a map from the interval [0, τ ] to the 2f dimensional phase space), whereas the 86 symbol Γ t will be used to denote the specific point in phase space visited by the trajectory Γ at time t.

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The central question is how the probability P [Γ] compares with the probability P [ Γ] to observe Γ, the 88 time-reversal companion of Γ: Γ t = εΓ τ −t where ε(q, p) = (q, −p) denotes the time reversal operator.

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The answer is given by the microreversibility principle which implies: To see this, consider the Hamiltonian dynamics but for the case that the trajectory Γ is not a solution

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To summarize, the micro reversibility principle for autonomous systems in conjunction with the hy-99 pothesis of Gibbsian equilibrium implies that the probability to observe a trajectory and its time-reversal companion are equal. There is no way to distinguish between past and future in an autonomous system 101 at equilibrium. Obviously, this is no longer so when the system is prepared out of equilibrium, as in Fig  assume that the varying parameter, denoted by λ t couples linearly to some system observable Q(q, p), 108 so that the Hamiltonian reads: This is the traditional form employed in the study of the fluctuation-dissipation theorem [13]. 4 In the 110 following we shall reserve the symbol λ (without subscript) to denote the whole parameter variation 111 protocol, and use the symbol λ t , to denote the specific value taken by the parameter at time t. The 112 succession of parameter values is assumed to be pre-specified (the system evolution does not affect the 113 parameter evolution). 114 We assume that λ t = 0 for t = 0 and that the system is prepared at t = 0 in the equilibrium Gibbs where Z 0 (β) = dqdqe −βH 0 (q,p) . We further assume that at any fixed value of the parameter the 117 Hamiltonian is time reversal symmetric: Note here the fact that energy is not conserved in the nonautonomous case because the Hamiltonian 119 is time-dependent in this case. Microreveresibility, as we have described it above, also does not hold:

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Given a protocol λ, if Γ is a solution of the Hamilton equations of motion, in general Γ is not. However

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Γ is a solution of the equations of motion generated by the time-reversed protocol λ, This is the microreversibility principle for nonautonmous systems [5]. It is illustrated in Fig. 2. Despite 123 its importance we are not aware of any text-books in classical (or quantum) mechanics that discusses it.

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A classical proof appears in [14, Sec.
Here, Q t = Q(Γ t ) denotes the evolution of the quantity Q along the trajectory Γ and W 0 is the so called  In order to see this it is important to realize that the work W 0 is odd under time-reversal. This is so Second Law (Fluctuation Theorem) Injecting some amount of energy W 0 into a thermally insulated system at equilibrium at temperature T by the cyclic variation of a parameter, is exponentially (i.e. 148 by a factor e W 0 /(k B T ) ) more probable than withdrawing the same amount of energy from it by the 149 reversed parameter variation.
The subscript λ in Eq. (9) is there to recall that the average is taken over the trajectories generated by the 152 protocol λ. In particular, the notation · λ denotes an nonequilibrium average. 6 Combining Eq. (9) with 153 Jensen's inequality, exp(x) ≥ exp( x ), leads to where Q is the time reversal companion of Q: Q t = Q τ −t . Now multiplying both sides of Eq. (12) by 167 Q τ and integrating over all Q-trajectories, one obtains: Note that Q τ λ = Q 0 λ and that, due to causality, the value taken by the observable Q(q, p) at time 169 t = 0 cannot be influenced by the subsequent evolution of the protocol λ. Therefore, the average presents 170 6 The nonequilibrium average · λ can be understood as an average over the work probability density function p[W 0 ; λ], that is the probability that the energy W 0 is injected in the system during one realization of the driving protocol. It formally reads[5]: p[W 0 ; λ] = dq 0 dp 0 ρ 0 (q 0 , p 0 )δ[W 0 − τ 0 λ tQ (q t , p t )], where δ denotes Dirac's delta function, and (q t , p t ) is the evolved of its initial (q 0 , p 0 ) under the driving protocol λ. a manifest equilibrium average; that is to say that it is an average over the initial canonical equilibrium 171 0 (q, p). We denote this equilibrium average by the symbol · (with no subscript). Thus, Eq. (13) reads 172 173 Q τ e −β λsQsds λ = Q 0 (14) By expanding the exponential in Eq. (14) to first order in λ, one obtains: Since the bracketed expression on the rhs is already O(λ) we can replace the non-equilibrium average 175 · λ with the equilibrium average · on the rhs. Further, since averaging commutes with time integration 176 one arrives, up to order O(λ 2 ), at: In the second line we made use of the time-homogeneous nature of the equilibrium correlation func-  Eq. (7).

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Just imagine we are shown a movie of an experiment in which a system starting at temperature T = (k B β) −1 is driven by a protocol, and we are asked to guess whether the movie is displayed in the same direction as it was filmed or in the backward direction, knowing that tossing of an unbiased coin decided the direction of the movie. When the outcome is +(−), the movie is shown in the same(opposite) direction as it was filmed. Imagine next that we can infer from the analysis of each single frame t the instantaneous values λ t and Q t taken by the parameter and its conjugate observable, respectively. With these we can evaluate the work W 0 for the displayed process using Eq. (8). Envision that we find, for the shown movie that βW 0 1. If the film was shown in the "correct" direction it means that a process corresponding to βW 0 1 occurred. If the film was shown backward then it means that a process corresponding to βW 0 −1 occurred (recall that W 0 is odd under time-reversal). The fluctuation theorem tells us that the former case occurs with an overwhelmingly higher probability relative to the Figure 3. Degree of belief P (+|W 0 ) that a movie showing the nonautonomous evolution of a system is shown in the same temporal order as it was filmed, given that the work W 0 was observed and that the direction of the movie was decided by the tossing of an unbiased coin.
probability of the latter case. Then we can be very much confident that the film was running in the correct direction. Likewise if we observe βW 0 −1, then we can say with very much confidence the the film depicts the process in the opposite direction as it happened. Clearly when intermediate values of βW 0 are observed we can still make well informed guesses about the direction of the movie, but with less confidence. The worst scenario arises when we observe W 0 = 0, in which case we cannot make any reliable guess. The question then arises of how to quantify the confidence of our guesses. This is a typical problem of Bayesian inference. Before we are shown the movie our degree of belief of the outcome +, is given by the prior, P (+) = 1/2 (likewise, P (−) = 1 − P (+) = 1/2). After we have seen the movie the prior is updated to the posterior, P (+|W 0 ), which is the degree of belief that the outcome + occurred, given the observed work W 0 . Using Bayes theorem, the posterior is given by where P (W 0 |+) is the conditional probability to observe W 0 given that + occurred, and P (W 0 ) is   The transition to certainty of guess occurs quite rapidly (in fact exponentially) around |βW 0 | 5. Note 199 that that for an autonomous system W 0 = 0, implying P (+|W 0 ) = P (−|W 0 ) = 1/2, meaning that, 200 as we have elaborated above, there is no way to discern the direction of time's arrow in an autonomous 201 system at equilibrium.

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Since the fluctuation theorem (7) holds as a general law regardless of the size of the system, it appears 203 that our ability to discern the direction of time's arrow does not depend on the system size. It is also worth mentioning the role played by thermal fluctuations in shaping our guesses. Particularly, with a 205 given observed value W 0 , the lower the temperature, the higher is the confidence (and vice-versa).