Fluctuation Theorems

Shiling Liang Institute of Physics, Ecole´ Polytechnique F´ed´erale de Lausanne (EPFL), 1015 Lausanne, Switzerland

This report is based on the review article by E.M Sevick et al.[1] on the ﬂuctuation theorem. I also refer to the original papers of Jarzynski [2–4] and adopt notation from a book written by Evans et al. [5]. I discuss the motivation for proposing the ﬂuctuation theorem and provide a detailed derivation of the Evans-Searles Fluctuation Theorem, which shows how macroscopic irreversibility emerged from the time-symmetrical equation of motion and addresses the importance of the initial condition of the system. I also recap the derivation the Crooks Fluctuation Theorem to relate work and free energy of a non-quasistatic process between two states of equilibrium, and I then take the Jarzynski equality as a consequence of the Crooks Fluctuation Theorem. In section III, it is shown that a simple physical example with an analytical solution can be used to verify these relations.

I. INTRODUCTION the second law of thermodynamics is a consequence of the fluctuation theorem. Fluctuation theorems present an analytical description of violations of the second law of thermodynamics; however, they also provide rigorous proof to this law from II. DERIVATION OF FLUCTUATION a statistical perspective. Classical thermodynamics can THEOREMS only deal with equilibrium systems and quasistatic processes. The behaviours of non-equilibrium systems are A. Phase space and Liouville’s Equation unclear within this scheme. Moreover, reversible equations of motion conflict with the monotonically increas- In Hamiltonian mechanics, all possible states of a sys- ing entropy predicted by the second law. Boltzmann’s tem expanded a phase space. A typical system in our statistical thermodynamics provides the groundwork for scale has more than ∼ 1023 particles. Thus, it is not solving this problem by using the second law as a statis- possible to know their exact configurations, so we use tical law. However, he was not able to determine a quan- a probability distribution function to characterise such titive relation. The most recent progress toward this end systems. By denoting the probability distribution func- was the development of the fluctuation theorem in the tion as f(Γ, t), the probability of finding the system in 1990s. an infinitesimal phase space volume dΓ is In 1994, Evans and Searles[6] proposed the fluctuation theorem, which calculates the probability of negative and positive entropy production. It is valid for P (dΓt, t) ≡ f(Γt, t)dΓt. (1) systems of any size and with arbitrary distances from equilibrium. It illustrates how irreversible macroscopic For a Hamiltonian system, the total time derivative of phenomena emerge from time-reversible equations of mo- f is tion. In 1997, Jarzynski[2] developed a more practical relation, Jarzynski equality, which allows us to establish df ∂f ∂f ∂f the relationships between work and free energy changes = + p˙ + q˙ dt ∂t ∂p ∂q for a non-quasistatic process between two states of equi- (2) ∂f ∂f librium. This relation reveals more information than = + · Γ˙ . do classical thermodynamics, which only reveals discrep- ∂t ∂Γ ancies between work and free energy differences. The next year, Crooks[7] developed a more general form of The continuity equation must hold for the probability the Jarzynski equality: the Crooks Fluctuation Theo- flow in the phase space rem. This theorem calculates the relative probability of given forward and backward trajectories. ∂(fΓ)˙ ∂f In this report, I recap the derivations of these relations + = 0. (3) ∂Γ ∂t based on a deterministic Hamiltonian system. I include a short introduction to phase spaces at the beginning Using these two equations to eliminate the partial deriva- and then follow the logical order rather than the chrono- tive of t, one obtains logical order of theorem development (which regards the Jarzynski equality as a consequence of the Crooks Fluc- df ∂Γ˙ = −f . (4) tuation Theorem). It is also demonstrated how to view dt ∂Γ 2

∂Γ˙ We now denote Λ ≡ ∂Γ , and integrate from the initial The positive dissipation function means the anti- time t = 0 to t = τ trajectories are less probable, and the negative dissipa- Z τ tion function gives the opposite case. It takes the value f(Γτ , τ) = f (Γ0, 0) exp − Λdt . (5) 0 only in the state of equilibrium, where no macroscopic 0 time evolution can be observed. Moreover, the dissipation function has odd parity As the probability of a speciﬁc phase space volume must be conserved, we can use equation 2 and equation ∗ 5 to obtain the compression factor for the inﬁnitesimal Ωτ (Γ0) = −Ωτ (Γ0). (11) phase space volume: Now it is important to calculate the probability of an dΓ Z τ anti-trajectory occurring within the entire phase space. τ = exp Λds . (6) dΓ0 0 We can do the following integral to calculate the relative probability of the dissipation function taking the opposite value. B. The Evans-Searles Fluctuation Theorem (1994)

p(Ωτ = A) The Evans-Searles fluctuation theorem is the most gen- p(Ωτ = −A) eral one, and it can be applied to a system of any size R dΓ δ(Ω (Γ ) − A)f(Γ , 0) and any distance from equilibrium. = 0 τ 0 0 t R ∗ ∗ ∗ Using time evolution operator S to denote the evo- dΓ0δ(Ωτ (Γ0) + A)f(Γ0, 0) R lution of a phase space point for a period t and time- dΓ0δ(Ωτ (Γ0) − A)f(Γ0, 0) (12) T reversal mapping operator M to reverse the trajectory. =R ∗ ∗ dΓ0δ(−Ωτ (Γ0) + A)f(Γ0, 0) For a Hamiltonian system, the reverse mapping is noting R but reversing the momentum p → −p. Since the dy- dΓ0δ(Ωτ (Γ0) − A)f(Γ0, 0) =R namics are deterministic, a trajectory can be determined dΓ0δ(−Ωτ (Γ0) + A)f(Γ0, 0) exp(−Ωτ (Γ0)) by its initial state. For a trajectory origin at phase point = exp(A), Γ0, the corresponding anti-trajectory can be easily repre- ∗ T T τ sented by its initial phase point Γ0 ≡ M Γt = M S Γ0. where we used the fact that Γ0 is a dummy variable for Note that the time-reversal mapping does not change the the integral over the whole phase space. Then we use the size of phase space volume. With equation (3), we can equation 11. To calculate the third equality, we used the obtain equation 7 with the definition of the dissipation function. The result, Z τ ∗ dΓ0 = dΓτ = dΓ0 exp Λdt . (7) p(Ωτ = A) 0 = exp(A), (13) p(Ωτ = −A) Alongside the trajectory of a single-phase space point, implies that for all trajectories in the phase space, irre- we also consider the evolution of an infinitesimal phase versible cases are exponentially more probable than are space volume dΓ ; that is, a bundle of trajectories. The 0 reversible cases. We can further calculate the average of probability of observing this volume is the dissipation function over the whole phase space:

∞ P (dΓ0, 0) ≡ dΓ0f(Γ0, 0). (8) Z hexp(−Ωτ )i = dA exp(−A)p(Ωτ = A) Macroscopic reversible systems indicate that the proba- −∞ Z ∞ bility of observing the forward bundle of trajectories and = dA exp(A)p(−Ωτ ) exp(A) the corresponding anti-trajectories should be the same: −∞ Z ∞ = dA p(Ωτ = −A) ∗ ∗ −∞ dΓ0f(Γ0, 0) = dΓ0f(Γ0, 0). (9) = 1. For more general cases, we deﬁne a dissipation function to characterize the reversibility of the bundle of trajectories Note that exp(−Ωτ ) is convex function, which lead to the Jensen’s inequality

P (dΓ , 0) Ω (Γ ) ≡ ln 0 τ 0 ∗ hΩτ i ≥ 0. (14) P (dΓ0, 0) Z τ (10) f(dΓ0, 0) This is exactly what we expect – the second law inequal- = ln ∗ − Λdt. f(dΓ0, 0) 0 ity. 3

C. Crooks Fluctuation Theorem (1998) Z τ ∂H(Γ, λ) ˙ ∂H(Γ, λ) ˙ Crooks Fluctuations Theorem and the Jarzynski equal- H(Γτ ,B) − H(Γ0,A) = dt λ + · Γ 0 ∂λ ∂Γ ity address the non-equilibrium processes between two (19) equilibriums (unlike classical thermodynamics, which can The first term of the integral is the total work done on only describe quasistatic work relations). Jarzynski the system, and the second term is the heat absorbed equality directly presents the work relation regarding ac- from the environment[8]. tions performed at any rate. The Crooks Fluctuation Theorem even measures the probability distribution of Z τ ∂H(Γ, λ) Z τ trajectories. These relations allow us to describe certain Q(Γ0, τ) ≡ dt · Γ˙ = kBT Λ(Γt)dt microscopic operations, such as the stretching of a poly- 0 ∂Γ 0 mer. The Crooks Fluctuation Theorem was developed later (20) than the Jarzynski equality. However, as it is a more gen- Z τ ∂H(Γ, λ) ˙ eral formula, we view the Jarzynski equality as a direct W (Γ0, τ) ≡ dt · λ. (21) ∂λ consequence of the Crooks Fluctuation Theorem in the 0 following discussion. The Crooks Fluctuation Theorem Hence, we can express the work done as a function of the reads energy difference associated with the initial and the final states, and we can express the phase space compression factor as pf (W = A) = exp (β(A − ∆F )) , (15) Z τ p (W = −A) r W (Γ0, τ) = H(Γτ ,B) − H(Γ0,A) − kBT Λ(Γt)dt. 0 where pf (W = A) is the probability of work done on the (22) system W = A between the initial equilibrium state A to As time evolution is deterministic for a Hamiltonian the final equilibrium state B. Additionally, pB(W = A) system, the probability distribution of trajectories with is the probability distribution of the reverse trajectories. W = A is directly determined by the initial distribution On the right-hand side, ∆F is the free energy difference function associated with the work parameter λ = A between the two states, and β ≡ 1 is the reverse tem- kB T Z perature of the heat bath. pf (W = A) = dΓ0δ(W (Γ0, τ) − A)p(H(Γ0,A)). (23) The external work is introduced by a control parameter λ. The Hamiltonian of the system can be written as In the same way, we can write the probability of the anti- trajectories during which the system does work A to the external agent: H(Γt, t, λt) = T (p) + V (q, λ), (16) where T and V are the kinetic energy and the poten- Z p (W = −A) = dΓ∗δ(W (Γ∗,A) + A)p(H(Γ∗,B)). tial, respectively. The potential can be adjusted by the b 0 0 0 external control parameter λ and varies from the initial (24) value λ = A to the final value λ = B. If it varies slowly Based on the results obtained, we can write the relative enough, it becomes a trivial quasistatic case. In more probability as general cases, an external agent drives the system out of equilibrium before it relaxes into a state of equilibrium p (W = A) once again. f p (W = −A) Initially, the system is in contact with a heat bath at a b R 1 dΓ0δ(W (Γ0, τ) − A)p(H(Γ0,A)) temperature of T = k β . Thus, it is a canonical ensemble B =R ∗ ∗ ∗ characterised by the Boltzmann probability distribution: dΓ0δ(W (Γ0, τ) + A)p(H(Γ0,B))

R −βH(Γ0,A) ZB dΓ0δ(W (Γ0, τ) − A)e = ∗ 1 R ∗ ∗ −βH(Γ0 ,B) −βH(Γ0,A) ZA dΓ0δ(W (Γ0, τ) + A)e (25) p(H(Γ0,A)) = e . (17) ZA Z R dΓ δ(W (Γ , τ) − A)e−βH(Γ0,A) = B 0 0 R −βH(Γ ,A) −βW (Γ ,τ) Then the λ varies from an initial value of A to final value ZA dΓ0δ(W (Γ0, τ) − A)e 0 e 0 of B in a specific protocol. The final equilibrium state ZB βA has the probability distribution = e ZA = exp (−β∆F + βA) , 1 p(H(Γ ,B)) = e−βH(Γτ ,B), (18) τ Z where the first equality is direct substitution of equation B 23 and 24. Then writing explicitly the equilibrium dis- and the change of Hamiltonian can be written as tribution leads to the second equality. To get the third 4 equality, we used equation 22 to rewrite the denominator. At time t = 0, we suddenly change the intensity of the With some simplification, we reach the final expression laser so that the spring constant jumps to k1. In this which is so-called Crooks fluctuation relation process, the work done on the system is dependent on p (W = A) the distance from the center at that moment f = exp (−β∆F + βA) . (26) pb(W = −A) Z τ ˙ dφext 1 2 W = dtλ = (k1 − k0)r0. (31) 0 dλ 2 D. Jarzynski equality (1997) Now the system is in an non-equilibrium state and will As we mentioned before, Jarzynski equality can be re- relax to a state of equilibrium: garded as a direct consequence of Crooks fluctuation the- 2π 1 2 −β 2 k1r orem. With this probability distribution of trajectories pk1 (r) = e . (32) βk given by Crooks FT, we can calculate the Jarzynski av- 1 erage of the work done on the system between initial and Accordingly, the free energy difference can be directly final state associated with λ = A and λ = B obtained from the initial and final equilibrium distribu- tions: Z k0 hexp(−βW )i = dA pf (W = A) exp(−βA) ∆F = −kBT ln Zk1 + kBT ln Zk0 = kBT ln (33) k1 Z (27) = dA pb(W = −A) exp(−β∆F ) To calculate the Jarzynski average of the work, we can first find the probability distribution of the trajectories = exp(−β∆F ). as a function of work by combining equation 30 and equa- This equality is highly useful; it allows for the calcula- tion 31: tion of the free energy difference by conducting the experiment many times. Due to the convexity of the expo- k1 − k0 −β k0W k1−k0 nential function, the Jarzynski equality implies pk0→k1 (W ) = pk0 (W ) = e . (34) βk0 Then, we can easily calculate the Jarzynski average and ∆F ≤ W. (28) confirm the Jarzynski equality: Therefore we can see it is consistent with the second law of thermodynamics. Z ∞ k W −βW −βW k1 − k0 −β 0 W he i = dW e e k1−k0 0 βk0 k1 (35) III. EXAMPLE = k0 A. Colloidal in an optical trap = e−β∆F . Following the same procedure, we can also calculate the The simplest and most practical example of applying probability of the reverse process p (W ) and confirm fluctuation theorems is through a single colloidal in an k0→k1 the Crooks Fluctuation Theorem: optical trap. This system has the smallest possible de- grees of freedom, and the optic trap can be approximated k W as a harmonical potential, which is the simplest poten- k −k −β 0 p (W ) 1 0 e k1−k0 tial providing restoring force. For such a system, we can k0→k1 βk0 β(W −∆F ) = k W = e . (36) k −k β 1 obtain an analytical solution to verify the fluctuation the- pk1→k0 (−W ) 0 1 k0−k1 βk e orems. 1 We treat the optical trap as a harmonical potential This example can also be used to verify the Evans- with the spring constant k and assume the particle con- Searles Fluctuation theorem. Since the Evans-Searles fined in the trap is over-damped; in this way, we can Fluctuation Theorem requires that all processes are time- neglect the kinetic energy. Thus, the Hamiltonian can reversible, we need to change the protocol to meet this be written as condition, which involves changing k1 back to k0 at time 1 t = τ. H(Γ , λ) = φ = k2r2, (29) 0 ext 2 where λ is the control parameter. At the beginning, k = IV. CONCLUSION k0 and the system is in a canonical distribution

2π 1 2 Fluctuation theorems bring us a deeper understand- −β 2 k0r pk0 (r) = e . (30) βk0 ing of the irreversible process. Evans-Searles fluctuations 5 resolved Lochimids objection on Boltzmann’s statistical without proof. For an intuitive understanding, we can thermodynamics. The Crooks Fluctuation Theorem and consider a specific case in which heat slowly enters in a Jarzynsky equality are powerful tools for investigating microcanonical ensemble. We begin by writing the above small systems with non-negligible fluctuations. However, relation in the instantaneous form: due to time restrictions, we can only discuss the deterministic dynamics described by Hamiltonian mechan- ∂H(Γ, λ) ics. The quantum version of the fluctuation theorem and Q˙ (Γt) ≡ · Γ˙ = kBT Λ(Γt) (38) stochastic dynamics are also worth discussing. ∂Γ To understand this, let us consider the case for states of equilibrium with quasistatic processes, in which Boltz- V. APPENDIX mann’s entropy is defined as

A. Hamiltonian mechanics with dissipative force S = kB log Ω, (39) See in section 2, Fluctuation Theorem, E.M Sevick et al.[1]. where Ω is the number of states. In a quasi-static process, we have

B. Heat absorbed and the phase space compression dQ dS = . (40) factor T

In the derivation of the Crooks Fluctuation Theorem, Thus, we used the relation dQ dS d Q˙ ≡ = T = kBT log Ω = kBT Λ(Γ). (41) Z τ ∂H(Γ, λ) Z τ dt dt dt Q(Γ0, τ) ≡ dt · Γ˙ = kBT Λ(Γt)dt (37) 0 ∂Γ 0

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