Nonequilibrim Fluctuation Theorems and Thermodynamic Second Laws

Hyunggyu Park School of Physics, KIAS

Abstract

Most processes in nature are nonequilibrium (NEQ) processes, which include key dynamic processes in biological cells and social networks, as well as the usual physical phenomena. Nevertheless, our understanding of NEQ dynamic processes, except for those near equilibrium (EQ) has been quite primitive. Recently, various interesting results have been reported regarding phenomena far from EQ. These have opened a new understanding of NEQ processes. Among many others, the most dramatic advance was achieved on the issue of the thermodynamic second law, which is known as the law of entropy increase or irreversibility. First, the inequality relation of the thermodynamic second law, S 0, was replaced by the equality relation, e-S =1, through which the inequality relation could be automatically derived. Detailed information on the probability distribution function of S was given by using the so-called Gallavotti-Cohen symmetry. More surprisingly, the total entropy production could be divided into two distinct terms, each of which satisfies its own thermodynamic 2nd law. This implies a more intricate structure in the irreversibility. In addition, many other new thermodynamic second laws and corresponding equality relations have been established. These equality relations are called the fluctuation theorems.

Prelude

Most processes in nature are non-equilibrium processes. The ‘equally likely postulate’ formed by Boltzmann and Gibbs a century ago can be applied only to equilibrium processes, and there is currently a lack of a general ensemble theorem that can describe irreversible dynamic processes from one equilibrium state to another, or a steady state that in itself is in non-equilibrium. However, many interesting processes, such as biological phenomena, are dynamic non-equilibrium processes far away from equilibrium. In particular, interest in

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non-equilibrium states has increased as experiments became incredibly elaborate on diverse dynamic processes in mesoscopic systems, such as motion of a molecular motor, dynamics of DNA molecules, chemical reactions of various protein molecules, and electron transport through quantum dots, and so on. Research interest of many statistical physicists has migrated from equilibrium problems to non-equilibrium problems, and the understanding of non-equilibrium processes has shown tremendous advances during the last 20 years. Establishment of the non-equilibrium fluctuation theorem has been the milestone of this advancement, and a drastic scientific advance has been achieved on the issue of the most mysterious law among the laws of nature; the second law of thermodynamics. A brief introduction of this recent progress is presented in this article [1].

Establishment of the non-equilibrium fluctuation theorem

Prior to a discussion on non-equilibrium, a distinction must be made between the equilibrium state and the non-equilibrium state. Equilibrium states are generally known as any state in which macroscopic quantities remain unchanged (of course, there are very complex and persistent microscopic fluctuations). However, a clearer definition is required for non-isolated systems. In dynamics, ‘detailed balance condition’ can be used for this purpose. When this condition is satisfied, the probability current from one microscopic state to another in a many-body system becomes balanced, and therefore, macroscopic quantities remain constant. However, the macroscopic quantities may not change even if the detailed balance condition is not satisfied. To distinguish it from the equilibrium state, we call this the non-equilibrium steady state. Simple examples of non-equilibrium steady states are water flow with a constant velocity, and constant electric current through a wire.

Both the general probability theory on non-equilibrium steady states and the universal microscopic theory on non-equilibrium dynamic states with time-dependent macroscopic quantities have not been established. The only phenomenological fact that has been established is that the entropy of a total (isolated) system has to increase in a non-equilibrium process: S 0 (in case of non-equilibrium steady states, the entropy is continuously produced even if macroscopic quantities do not change). This is called ‘the second law of thermodynamics’. Since this law defines the direction of time (“arrow of time”), it is very difficult to understand this law from microscopic theories with the time-reversal symmetry, such as quantum mechanics and relativity. Efforts to understand this law have persisted for a long time, but they have not yet been successful at the fundamental level.

Since 1993, new and remarkable results beyond the second law of thermodynamics have been reported. The initial discovery was serendipitous. Using molecular dynamics computer simulations, Evans, Cohen, and Morris have numerically observed that a fluid system exhibited an interesting symmetry in the probability distribution of the entropy production in the long-term non-equilibrium process [2].

P(S) = eS P(-S)

THE KIAS Newsletter 2013 5 On average, the entropy production is positive ( S 0), but in some samples, negative cases (S<0) also exist, albeit with an exponentially lower probability than that of positive cases. It is not surprising that the probability of negative cases was exponentially smaller. And in the long-time limit or in the thermodynamic limit, the probability of S<0 cases becomes extremely small, so this case cannot be observed in real experiments on macroscopic systems. However, in a mesoscopic system, it can still be observed at a low probability. In fact, numerous experimental observations have been reported in various mesoscopic systems during the last ten years. However, these experimental reports must not be misunderstood as a violation of the second law of thermodynamics. It has long been predicted that S<0 cases will be observed unless the system is not in the thermodynamic limit, although there was no precise prediction for its probability. The equation above is the prediction for that probability.

Theoretically, it is easy to predict that the ratio of the probabilities would be exponential. Therefore, the importance of the equation above is not that the exponent of the exponential function is proportional to S but that the proportional constant is exactly 1. This equation has been analytically derived by Evans and Searles, and Gallavotti and Cohen [3,4], and became the basis of the similar equations derived later on. These equations are collectively referred to as the “non-equilibrium fluctuation theorem,” and the positive-negative symmetry of this equation, in particular, is called the Gallavotti-Cohen symmetry. The reason why the word “fluctuation” is in its name is presumably because it gives the information about the probability distribution (fluctuation) not about the mean value .

This fluctuation theorem has been analytically proven to be valid within Langevin systems [5] and Markov

FIG. 1 Maxwell’s demon also makes mistakes. How often does he do? Detailed fluctuation theorems reveal its probability exactly. This figure is taken from http://www.kawai.phy.uab.edu/ktaog2/index.html

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processes [6], which are more general stochastic systems. This fluctuation theorem is also referred to as a ‘detailed’ fluctuation theorem, and can be proved to be valid even for a finite time interval if the system starts from a steady state(transient fluctuation systems).

Integral fluctuation theorem and the second law of thermodynamics

The following equation can be easily derived from the detailed fluctuation theorem:

e-S =d(S) P(S) e-S =1

It is called an integral fluctuation theorem. The interesting fact is that the second law of thermodynamics ( S 0) can be easily derived from the integral fluctuation theorem, using the Jensen’s inequality ( e x e x ). Therefore, one may say that the integral fluctuation theorem is more fundamental than the second law of thermodynamics. Since this theorem is given by an equation rather than an inequality, its precision can be directly tested against real experimental data. In summary, the integral fluctuation theorem, or on a more fundamental level, the detailed fluctuation theorem is the most fundamental law which breaks the time-reversal symmetry. The second law of thermodynamics is just a derivative of these fundamental theorems. These theorems give the explicit symmetric information on the probability distribution (fluctuation) of the entropy production.

Work, free energy, and Jarzynski identity

In 1997, Jarzynski derived another fluctuation theorem on a Hamiltonian system using just a few lines of calculations [7]. This theorem has successfully related the free energy, a representative equilibrium physical quantity, with the probability function of work, which is representative of non-equilibrium physical quantities along with the entropy production and heat. This integral fluctuation theorem, referred to as the Jarzynski identity, may be written as follows:

e -W = e -F

Here, W is the Jarzynski work done during a non-equilibrium process, F is the change in the free energy and is the inverse of the temperature. For a more detailed definition, refer to [7]. In order for this integral fluctuation theorem to be valid, a special initial condition is reguired, i.e., the system must initially be in equilibrium. In contrast, the integral fluctuation theorem about the entropy production does not require this special initial condition.

It is natural to maintain an equilibrium state initially in real experiments. Therefore, many experiments measured the work, which is much easier to be measured than entropy, and tested the above work-free energy fluctuation theorem. In addition, the detailed fluctuation theorem has been derived similar to the entropy production (Crooks

THE KIAS Newsletter 2013 7 detailed fluctuation theorem) [8]. These work-free energy fluctuation theorems have been shown to be valid even in more general stochastic systems. The first related experiment was the DNA unfolding experiment published in Nature 2005 [9], which confirmed that both the detailed and the integral fluctuation theorems are indeed working well. Recently, more precise verifications are being realized through electron-transport experiments using quantum dots.

Independent second laws of thermodynamics and entropies

Coming into the 21st century, some more interesting and surprising fluctuation theorems have been discovered. Hatano and Sasa revealed that the entropy production can be divided into two parts according to its physical origin, in particular, showing that the excess entropy, which is a part of the entropy related to excess heat, satisfies the integral fluctuation theorem by itself [10]. Subsequently, Speck and Seifert showed that the other entropy, the ‘housekeeping entropy’, satisfies an independent integral fluctuation theorem [11]. These observations show that the well-known second law of thermodynamics, which states that “the entropy always increases” is actually a result of independent increases of the excess entropy and the housekeeping entropy. This opens a possibility that for more complex systems, there may be multiple fundamental quantities that break the time-reversal symmetry and endow time with the direction. The situation is analogous to the question of how many independent laws of conservation exist in a mechanical system. Many research groups around the globe, including those in Korea, are actively studying this subject.

Epilog

Fluctuation theorems discussed above may not be valid in all non-equilibrium processes, but can still be applied to a wide range of processes. Recently, many statistical physicists are searching for the validity range of these fluctuation theorems, and are trying to discover new fluctuation theorems and their physical origin. In particular, recent researches are focused on the effect of information entropy involved with the Maxwell’s Demon and also odd-parity variables under the time reversal such as momentum or spin on these fluctuation theorems.

It is hard to predict how influential these fluctuation theorems will be on the understanding of non-equilibrium systems in general, but to be sure, these new fundamental discoveries have brought us closer to the unprecedented understanding of non-equilibrium processes. This may serve as the first step of an ultimate solution to understand origin, maintenance, and evolution of life which is the most typical example of non-equilibrium phenomena. With recent rapid progress, we can be optimistic about finding a hint toward the solution of the problem before the end of the 21st century. Lastly, some recent review articles are introduced below for those who are interested [12-14].

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Reference

[1] H. Park, “Advances in statistical mechanics and Recent trends”, KIAS Newsletter 44, 6 (2011). [2] D. J. Evans, E. G. D. Cohen and G. P. Morris, Phys. Rev. Lett. 71, 2041 (1993). [3] D. J. Evans and D. J. Searles, Phys. Rev. E 50, 1645 (1994). [4] G. Gallavotti and E. G. D. Cohen, J. Stat. Phys. 80, 931 (1995). [5] J. Kurchan, J. Phys. A 31, 3719 (1998). [6] J. L. Lebowitz and H. Spohn, J. Stat. Phys. 95, 333 (1999). [7] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997). [8] G. E. Crooks, J. Stat. Phys. 90, 1481 (1998). [9] Collin et al., Nature 437, 8 (2005). [10] T. Hatano and S. I. Sasa, Phys. Rev. Lett. 86, 3463 (2001). [11] T. Speck and U. Seifert, J. Phys. A 38, L581 (2005). [12] C. Jarzynski, Annu. Rev. Condens. Matter Phys. 2, 329 (2011). [13] J. Kurchan, J. Stat. Mech. P07005 (2007). [14] U. Seifert, Eur. Phys. J. B 64, 423 (2008). * This article was published in the Physics and High Technology, Volume 21, Issue 21, pp 32-33, December 2012 (The Korean Physical Society) in Korean. It was translated into English and published in the KIAS Newsletter under the permission of “Physics and High Technology”.

Hyunggyu Park

Hyunggyu Park is a Professor at the School of Physics since 2002. He has a broad experience in investigating emergent properties of general many body systems, ranging from classical magnetic systems, incommensurate systems, and quasicrystals to nonequilibrium stochastic complex systems.

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