Fluctuations and large deviations in non-equilibrium systems

Abstract

Equilibrium thermodynamics is one of the successful theories of all time. It encapsulates the phenomena of large macroscopic systems which are in thermal equilibrium. When two systems are in equilibrium with each other, and a third system is brought in contact with either of these two by a diathermal wall, then, all of them will reach in an equilibrium state. This principle is referred to as the zeroth law of thermodynamics. As a consequence, it helps to define an empirical temperature such that systems are in an equilibrium state will have the same temperature. In terms of conservation of energy, the first law of thermodynamics states that change in the internal energy is the sum of the amount of work done on the system and the heat supplied to the system. There are several processes which obey the first law of thermodynamics, but they do not occur in nature. For example, heat never flows from a cold body to a hot body. The second law of thermodynamics incorporates such observations in the thermodynamics and defines a state function called entropy which characterizes the irreversibility of a process. For an isolated system in the equilibrium, the entropy attains a maximum value. These laws of thermodynamics are well understood from the microscopic degrees of freedom of a system in a well-established framework known as equilibrium statistical mechanics. In this context, the probability of the system to find in a given configuration is expressed as Gibbs-Boltzmann weight. Moreover, the macroscopic properties can be evaluated, in principle, from a normalization constant called the partition function. In contrast to equilibrium picture, non-equilibrium phenomena cover a much larger class of problems in science. These phenomena can be seen in biology, chemistry, physics, mathematics, ecology, financial markets, etc. Because of driving fields such as temperature or chemical potential gradient, shear flow, external time-dependent fields, etc., these systems are not in the equilibrium state. Such driving fields are known as affinity or generalized force. There is no such general theory present which describes the methodology to study the observable properties of the system away from equilibrium. Nevertheless, linear irreversible thermodynamics is a very helpful tool to understand the physics of systems which xiv are close to equilibrium. The foundation of linear irreversible thermodynamics is based on time-reversal invariance principle and the postulates of equilibrium thermodynamics. Within this context, one can write the rate of entropy production as the sum of the product of each flux with its associated affinity (generalized force). For purely resistive systems, each local flux depends upon instantaneous values of all affinities. If affinities are so small, each local flux is related to all of them through kinetic coefficients linearly. These kinetic coefficients depend upon the local intensive parameters of the system. Moreover, in the linear regime, one can study Thomson effect, Peltier effect, Seebeck effect, etc. In the case of non-purely resistive systems, local flux depends upon the value of affinities at the instantaneous as well as the previous times, i.e. systems have memory. Here, one can study the properties of the system using the fluctuation-dissipation theorem or Kubo formula. When these systems are far away from equilibrium, there is no such non-equilibrium counterpart of the equilibrium partition function with which one can compute the observable properties. This thesis compiles a set of physical problems where the systems under observation are in microscopic scale. These small systems can be biopolymers (like DNA, RNA, protein molecules), enzymes, Brownian particle, Brownian motors, small scale heat engine, small electronic systems, etc. As the system size reduces, fluctuations present in the surrounding bath perturb the deterministic behaviour of the system. Nevertheless, the probabilistic nature of these small systems can be exactly understood by Fokker-Planck or master equation. These equations play an essential role to write down the evolution of the probability of each configuration visited by the system. For a generic irreversible process, work done on the system is bounded below by the change in free energy of the initial and final states. Similar bound one can see for the entropy change between two states. Fluctuation relations extend these bounds into exact equalities. These identities are proven to be remarkable results in the area of non-equilibrium statistical mechanics. For example, Jarzynki’s equality helps in the computation of free energy difference between two equilibrium states using work done in a non-equilibrium protocol. Fluctuation theorem relates the xv probabilities of positive and negative entropy production in both steady and transient state. Earlier, it was assumed that the total entropy production is only due to the heat exchange by the system to the environment (medium entropy production) in a non-equilibrium process. This entropy production is found to be the ratio of path probabilities of the forward to that of the reverse process when the dynamics of the system is microscopically reversible. But later, it was found that the fluctuation theorem for total entropy production in the steady state is valid once we incorporate the entropy production of the system to the total entropy production. The entropy production of the system is called entropy change which occurs due to configurational change between the initial and final states. This theorem is independent of the driving protocol and is valid for all time. While fluctuation relations display exciting results in the realm of non-equilibrium statistical physics, they do not give the insight of the individual probability distribution of stochastic quantities such as work done, heat flow, entropy, efficiency, etc. In particular, one is also interested in the large time statistics of these observables. In this thesis, we will focus on both the fluctuation theorem as well as the probability density of these stochastic quantities. Fluctuation theorem for partial entropy production Consider a system where a large number of degrees of freedom are interacting with each other. This given system is connected to a heat bath of constant temperature. Evidently, the total entropy production is zero, i.e. the system is in equilibrium. Suppose the whole system is driven by some external forces (these forces are explained above), then the total entropy production is not zero. Therefore, the total entropy production characterizes the non-equilibrium system and the measurement of it is quite essential. In the steady state, it satisfies the fluctuation theorem as mentioned above. In some experiments, it is possible to have some setup where entropy production is to be observed. In such settings, we have to understand the clear time-scale separations among the degrees of freedom. Based on time-scales of relaxation, we classify degrees of freedoms as fast or slow variables. In some earlier experimental xvi and theoretical studies, it was observed that the fluctuation theorem for total entropy production would hold once we observe all slow degrees of freedom. If one may not succeed to observe all relevant slow degrees of freedom of the system, the exact estimate of the total entropy production would be difficult. Thus, the fluctuation theorem for the total entropy production based on partial information may not hold. In some earlier studies, the order of violation of fluctuation theorem was observed to be proportional to the interaction parameter between the observed degrees of freedom and the hidden ones (which we cannot observe). When such an interaction parameter is taken to be very small called the weak coupling limit, the fluctuation theorem restores its form which is also a naive guess one can think of. On the contrary to that, in this thesis, we have considered some simple analytically tractable model systems where we estimate the violation of the fluctuation theorem even in the weak coupling limit. Therefore, it is essential to consider the hidden degrees of freedom even though these are weakly coupled to the observed ones. Moreover, we have given a recipe with which one can nullify the effect of the weak coupling of the hidden variables on the observed ones. A given system A is coupled to another system B of the same kind (both systems have same time-scale of relaxation), jointly called system C. The system C is in contact with the heat bath of a constant temperature. External forces (these forces can be correlated as well as uncorrelated with each other) are used to drive system C out of equilibrium and entropy is generated. Total entropy production by the coupled system C satisfies the fluctuation theorem in the steady state. The entropy production due to system A in the coupled system C is computed and shown that this entropy production does not obey steady state fluctuation theorem even in weak coupling limit which is a remarkable result. In one more setup, we have considered two Brownian particles (A and B) coupled via harmonic interaction. One of the particles (say A) is connected to a temperature gradient while the other one B is connected to one heat bath of a constant temperature. In general, all of these heat baths do not have same temperature. In the steady state, we compute the entropy production due to xvii particle A in the coupled system and show that it does not satisfy the fluctuation theorem in weak coupling limit. Consider two Brownian particles A and B and these particles are coupled by harmonic interaction. Both particles are confined in a harmonic trap. The whole system is immersed in the heat bath of a constant temperature. Both of these particles are driven by some external forces. It is shown that the entropy production by one of the particles in the coupled system in steady state, satisfies the fluctuation theorem in weak coupling limit. Suppose a Brownian particle is connected to two heat baths (B1 and B2) of different temperatures. Then, we couple the given particle to one more heat bath B3 of a distinct temperature weakly. The observable is the total entropy production by the particle due to baths B1 and B2 in the steady state, and we show that it satisfies the fluctuation theorem when bath B3 is weakly coupled to the particle. Stochastic efficiency of an isothermal work-to-work converter engine In thermodynamics, a heat engine is a machine used to do work by consuming energy in a cyclic process. The efficiency of any macroscopic engine (working substance 1023 particles) is limited by the Carnot’s theorem. This bound is universal and does not depend on the nature of constituents of the engine. Recently, there has been a lot of research going on to investigate the nature of these heat engines in the microscopic scale. On this scale, thermal fluctuation predominates, and hence, observables such as work done, heat flow, entropy, efficiency, etc. become stochastic quantity. Therefore, the probability density function of the efficiency (ratio of output power to the input power) of a microscopic heat engine becomes a good candidate to examine. In this thesis, we extend this study by giving two simple model systems of such a microscopic engine which converts the input work to the output work. xviii A Brownian particle is immersed in a heat bath of a constant temperature. For this system to function as an engine called isothermal work-to-work converter engine, we apply stochastic external forces. These forces are called drive and load forces. The function of the drive force is to drive the particle against the load force. Since the system is in the microscopic regime, work done by these forces are random variables. The efficiency of such engine defined as the ratio of output work to the input work, is also a stochastic variable. Using this setting, we analytically obtain the large but finite time probability density function of the stochastic efficiency. Consider an isothermal machine composed of two Brownian particles (say particle A and B) connected by a harmonic spring. A constant load is attached to particle A, and particle B is trapped in harmonic confinement whose minimum is dragged with a constant velocity. The distribution of work done on particle A, particle B, and on both particles together is obtained, and the corresponding transient fluctuation theorem is tested. Furthermore, the probability density function for the stochastic efficiency (output work/input work) of this machine is computed for all time. Briefly, in this thesis, we have explored the problems where the system size is very small and is driven away from the equilibrium using some driving protocol. In the steady state (except Chapter 7), we computed the large deviation function as well as the probability density function of stochastic observables such as work done, heat flow, entropy, efficiency, etc. In Chapter 7, the probability density functions for work done and stochastic efficiency are studied in the transient regime.