Session T52: Non-equilibrium Systems, Especially using Fluctuation Theorems and Fluctuation-Dissipation Relations

Transient and steady state behavior of full counting statistics in thermal transport

We study the statistics of heat transferred in a given time interval $t_M$, through a finite harmonic system, which is connected with two heat baths, maintained at two different temperatures. We calculate the cumulant generating function (CGF) for heat transfer using non-equilibrium Green's function method. The CGF can be concisely expressed in terms of Green's functions of the system and the self-energy of the lead with shifted arguments, $\Sigma^A(\tau, \tau') = \Sigma_L\bigl(\tau +\hbar x(\tau), \tau' + \hbar x(\tau')\bigr) - \Sigma_L(\tau, \tau')$, where $\Sigma_L(\tau,\tau')$ is the contour-ordered self-energy of the left lead. The expression of CGF is valid in both transient and steady state regimes. We present a transient result of the first four cumulants of a graphene junction. It is found that measurement causes the energy to flow into the leads. In the steady state we show that the CGF obey {\it``steady state fluctuation theorem''}. We also study the CGF for the joint probability distribution of left and right lead heat flux $P(Q_L,Q_R)$, which is important to calculate the correlations between $Q_L$ and $Q_R$, and also the total entropy that flows into the leads. We also discuss the CGF for the total entropy production for two lead system without the center part.   

Langevin dynamics beyond the weak coupling limit

Many popular results of non-equilibrium statistical mechanics hold only in leading order in a small parameter $\lambda$ which controls the strength of the system-environment coupling. In this approximation the equations for the first two moments $\langle v\rangle$ and $\langle v^2\rangle$ of the Brownian particle's velocity are closed and describe exponential relaxation to thermal equilibrium. To higher orders in $\lambda$ these equations are not closed but coupled to higher moments $\langle v^n\rangle$. This may result in much richer dynamics (both transient and stationary) and non-trivial ergodic properties. Generalized fluctuation-dissipation relations are derived microscopically and shown to ensure convergence to thermal equilibrium to any order in $\lambda$. One exception is the regime of superlinear diffusion, characterized by zero integral friction (vanishing integral of the memory kernel), when the generalized Langevin equation may have non-ergodic solutions that do not relax to equilibrium values. Also, for specific memory kernels the equation may have non-dissipative (non-stationary) solutions even if the integral friction is finite and diffusion is normal.   

Jarzynski equality for spin glasses and its application

We study an application of Jarzynski equality to spin glasses with gauge symmetry. It is shown that the exponentiated free-energy difference appearing in the Jarzynski equality reduces to a simple analytic function written explicitly in terms of the initial and final temperatures if the temperature satisfies a certain condition related to gauge symmetry. This result can be used to derive a lower bound on the performed work during the nonequilibrium process by changing the external magnetic field as well as a pseudo work done during changing temperature. The latter case serves as useful information to implement the population annealing developed in numerical use of the Jarzynski equality to equilibrate the many-body system. We also prove several exact identities that relate equilibrium and nonequilibrium quantities. These identities show possibility of the population annealing to evaluate equilibrium quantities from nonequilibrium computations, which may be useful for avoiding the problem of slow relaxation in spin glasses.   

Thermodynamic reversibility in feedback processes

The information acquired during a thermodynamic process with feedback can be converted into useful work. However, the second law of feedback restricts the amount of useful work that can be obtained from this information. In this presentation, I will discuss optimal thermodynamic processes with feedback, where all the information is converted into work. Utilizing the detailed fluctuation theorem for feedback, I will demonstrate that such processes are feedback-reversible: they are indistinguishable from their time-reversal, thereby extending the notion of thermodynamic reversibility to feedback processes.   

Entropy production in non-equilibrium steady states

We discuss entropy production in non-equilibrium steady states by focusing on paths obtained by sampling at regular (small) intervals, instead of sampling on each change of the system's state. This allows us to directly study entropy production in systems with microscopic irreversibility. The two sampling methods are equivalent otherwise, and the fluctuation theorem also holds for the different paths. We focus on a fully irreversible three-state loop, as a canonical model of microscopic irreversibility, finding its entropy distribution, rate of entropy production, and large deviation function in closed analytical form, and showing that the observed kink in the large deviation function arises solely from microscopic irreversibility [1].\\[4pt] [1] D. ben-Avraham, S. Dorosz, and M. Pleimling, Phys. Rev. E 84, 011115 (2011)   

Information, entropy, and heat far from equilibrium

For equilibrium states, the Shannon entropy, $H =- \int d\Gamma \,p(\Gamma) \ln p(\Gamma) $, coincides with the thermodynamic entropy. It has lately been recognized that for systems in nonequilibrium steady states a thermodynamic description based on $H$ becomes feasible, as well. In the present work we derive various generalizations of the second law for nonequilibrium processes with additional information supplied from an external or an internal memory. We will show that the irreversible entropy production is bounded from below by the information transferred from the memory to the system.   

Effective temperature determines density distribution in a slowly varying external potential beyond linear response

We consider a sheared colloidal suspension under the influence of an external potential that varies slowly in space in the plane perpendicular to the flow and acts on one selected (tagged) particle of the suspension. Using a Chapman-Enskog type expansion we derive a steady state equation for the tagged particle density distribution. We show that for potentials varying along one direction only, the tagged particle distribution is the same as the equilibrium distribution with the temperature equal to the effective temperature obtained from the violation of the Einstein relation between the self-diffusion and tagged particle mobility coefficients. We thus prove the usefulness of this effective temperature for the description of the tagged particle behavior beyond the realm of linear response. We illustrate our theoretical predictions with Brownian dynamics computer simulations.   

Dynamic phase transitions in large work production of linear diffusion systems

We present the theoretical study on non-equilibrium (NEQ) fluctuations for diffusion dynamics in high dimensions driven by a linear drift force. We find the time-dependent probability distribution function exactly as well as the NEQ work production distribution P(W) in terms of solutions of nonlinear differential equations. In two dimensions, we find analytically a sequence of dynamic phase transitions in the exponential tail shape of P(W). Their implications and orgins are discussed.   

Driven Langevin dynamics: heat, work and pseudo-work

Common algorithms for simulating Langevin dynamics are neither microscopically reversible, nor do they preserve the equilibrium distribution. Instead, even with a time-independent Hamiltonian, finite time step Langevin integrators model a driven, nonequilibrium dynamics that breaks time-reversal symmetry. Herein, we demonstrate that these problems can be resolved with a Langevin integrator that splits the dynamics into separate deterministic and stochastic substeps. This allows the total energy change of a driven system to be divided into heat, work, and pseudo-work -- the work induced by the finite time step. The extent of time-symmetry breaking due to the finite time step can be measured and true equilibrium properties recovered. This interpretation of discrete time step Langevin dynamics as a driven process provides new insights into the practical use of stochastic integrators for molecular simulation.   

Statistical interpretation of heat and work and the adiabatic theorem in irreversible processes

By generalizing the traditional concept [1] of heat and work to include their irreversible components allows us to express them in statistical terms so that dW is the \textit{isentropic} energy change; this generalizes the equilibrium adiabatic theorem. Then dQ is then nothing but the energy change solely due to the change in the microstate probability. Accordingly dQ=TdS [2] so that \textit{all powerful aspects of the equilibrium formulation are preserved}, a quite remarkable but unexpected result. The traditional formulation of the first law of thermodynamics, which uses the fields (temperature, pressure, etc.) of the medium can be \textit{equivalently} written as dE=dQ-dW using the fields of the system. This makes the two descriptions using the fields of the medium or the system equivalent and settles the long existing dispute in the literature regarding the proper choice of the fields. Moreover, the use of system fields (including affinities) allows us to analyze non-equilibrium processes such as free expansion between non-equilibrium states, which cannot be analyzed in the traditional approach. \\[4pt] [1] P.D. Gujrati, arXiv:1105.5549.\\[0pt] [2] P.D. Gujrati, Phys. Rev. E \textbf{81}, 051130 (2010).   

Application of the generalized fluctuation-dissipation theorem on a sheared suspension

We explore the validity of the generalized fluctuation-dissipation theorem for steady-state systems (proposed by Prost, Joanny and Parrondo, PRL 103, 090601 2009) in an experimental system: a suspension of non-colloidal spheres under slow periodic strain. The system is out-of-equilibrium and typically undergoes a phase transition from an active fluctuating to an absorbing state as the strain amplitude is decreased. It is a good candidate for applying the proposed theory since it has Markovian dynamics and fluctuating steady states. The control parameters are the applied strain amplitude and its volume fraction and fluctuations of proper observables such as the individual particle locations can be readily measured. Perturbations of the control parameter of strain can lead in new steady states after a transient response, which in turn can be correlated with the fluctuating observable, thus providing a way of verifying the validity of the proposed version of the generalized fluctuation-dissipation theorem.   

Entropy rate of non-equilibrium growing networks

In order to quantify the complexity of networks, new entropy measures have recently been introduced. Most of these entropy measures pertain to static networks or to dynamical processes defined on static complex networks. In this talk, we will discuss the entropy rate of growing network models, which quantifies how many labeled networks are typically generated by those growing network models. We will present an analytical evaluation of the difference between the entropy rate of growing tree network models and the entropy of tree networks that have the same asymptotic degree distribution. We will outline our finding that growing networks with linear preferential attachment generated by dynamical models are exponentially less in number than the static networks with the same degree distribution for a large variety of relevant growing network models. We will also discuss the entropy rate for growing network models that show structural phase transitions, including models with non-linear preferential attachment. We will conclude by presenting numerical simulations showing that the entropy rates above and below the structural phase transitions follow a different scaling with the network size.   

Persistency and Uncertainty Across the Academic Career

Recent shifts in the business structure of universities and a bottleneck in the supply of tenure track positions are two issues that threaten to change the longstanding patronage system in academia and affect the overall potential of science. We analyze the longitudinal publication rate $n_{i}(t)$ on the 1-year scale for 300 physicists $i=1...300$. For most careers analyzed, we observe cumulative production acceleration $N_{i}(t) \approx A_{i} t^{\alpha_{i}}$ with $\alpha_{i} >1$, reflecting the benefits of learning and collaboration spillovers which constitute a cumulative advantage. We find that the variance in production scales with collaboration radius size $S_{i}$ as $\sigma^{2}_{i} \sim S_{i}^{\psi}$ with $0.4 < \psi < 0.8$. We develop a preferential growth model to gain insight into the relation between career persistency and career uncertainty. This model shows that emphasis on nonstop production, a consequence of short-term contract systems, results in a significant number of ``sudden death'' careers that terminate due to unavoidable negative production shocks. Hence, short-term contracts may increase the strength of ``rich-get-richer'' mechanisms in competitive professions and hinder the upward mobility of young scientists.   

A Random Walk Picture of Basketball

We analyze NBA basketball play-by-play data and found that scoring is well described by a weakly-biased, anti-persistent, continuous-time random walk. The time between successive scoring events follows an exponential distribution, with little memory between events. We account for a wide variety of statistical properties of scoring, such as the distribution of the score difference between opponents and the fraction of game time that one team is in the lead.