Session A40: Systems Far From Equilibrium

Jarzynski equality for non-Hamiltonian dynamics

Recent years have witnessed major advances in our understanding of nonequilibrium processes. The Jarzynski equality, for example, provides a link between equilibrium free energy differences and finite-time, nonequilibrium dynamics. We propose a generalization of this relation to non-Hamiltonian dynamics, relevant for active matter systems, continuous feedback, and computer simulation. Surprisingly, this relation allows us to calculate the free energy difference between the desired initial and final states using arbitrary dynamics. As a practical matter, this dissociation between the dynamics and the initial and final states promises to facilitate a range of techniques for free energy estimation in a single, universal expression.   

Field Theoretic Description of Nonequilibrium Work Relations

We develop Doi-Peliti field theory for driven, interacting particles coupled to a thermal bath. This mapping of classical particles to a field theory does not require any assumption of large particle numbers or slow modes. As an application we consider nonequilibrium work relations. With the introduction of an auxiliary complex field, the Jarzynski relation emerges from the field theory as direct consequence of time reversal.   

Extending Landauer's Bound from Bit Erasure to Arbitrary Computation

Recent analyses have calculated the minimal thermodynamic work required to perform any computation $\pi$ whose output is independent of its input, e.g., bit erasure. First I extend these analyses to calculate the work required even if the output of $\pi$ depends on its input. Next I show that if a physical computer $\mathcal{C}$ implementing a computation $\pi$ will be re-used, then the work required depends only on the dynamics of the logical variables under $\pi$, independent of the physical details of $\mathcal{C}$. This establishes a formal identity between the thermodynamics of (re-usable) computers and theoretical computer science. To illustrate this identity, I prove that the minimal work required to compute a bit string $\sigma$ on a (physical) Turing machine $M$ is $k_BT\ln(2) \big[$Kolmogorov complexity($\sigma) \;+$ log (Bernoulli measure of the set of strings that compute $\sigma) \;+ $ log(halting probability of $M)\big]$. I also prove that uncertainty about the distribution over inputs to the computer increases the minimal work required to run the computer. I end by using these results to relate the free energy flux incident on an organism / robot / biosphere to the maximal amount of computation that the organism / robot / biosphere can do per unit time.   

Average work measurement below the Landauer limit for memory erasure

Landauer's Principle states that erasing a one-bit memory requires an average work of at least $kT\ln$2. Recent experiments have confirmed this prediction for a one-bit memory represented by a symmetric double-well potential. However, if a memory is represented by a non-equilibrium state in an asymmetric double-well potential, theoretical studies predict that one can measure work below $kT\ln$2. Using a feedback trap, we have confirmed this prediction. Surprisingly, we found that two different erasure protocols give two different values for the asymptotic work. We can explain this result by noting that one protocol is symmetric with the respect to time reversal and the other is not.   

Dynamics and thermodynamics of open chemical networks

Open chemical networks (OCN) are large sets of coupled chemical reactions where some of the species are chemostated (i.e. continuously restored from the environment). Cell metabolism is a notable example of OCN. Two results will be presented. First, dissipation in OCN operating in nonequilibrium steady-states strongly depends on the network topology (algebraic properties of the stoichiometric matrix) \footnote{M. Polettini and M. Esposito, J. Chem. Phys. 141, 024117 (2014)}. An application to oligosaccharides exchange dynamics performed by so-called D-enzymes will be provided \footnote{R. Rao, D. Lacoste and M. Esposito, arxiv:1509.07446}. Second, at low concentration the dissipation of OCN is in general inaccurately predicted by deterministic dynamics (i.e. nonlinear rate equations for the species concentrations). In this case a description in terms of the chemical master equation is necessary. A notable exception is provided by so-called deficiency zero networks, i.e. chemical networks with no hidden cycles present in the graph of reactant complexes \footnote{M. Polettini, A. Wachtel and M. Esposito, arxiv:1507.00058}.   

Cost and consequences of breaking the fluctuation dissipation relation in biochemical networks

Living systems need to be highly responsive, and also to keep fluctuations low. These goals are incompatible in equilibrium systems due to the Fluctuation Dissipation Theorem (FDT). Here, we show that biological sensory systems, driven far from equilibrium by free energy consumption, can reduce their intrinsic fluctuations while maintaining high responsiveness. By developing a continuum theory of the \textit{E. coli} chemotaxis pathway, we demonstrate that adaptation can be understood as a non-equilibrium phase transition controlled by free energy dissipation, and it is characterized by a breaking of the FDT [1]. We show that the maximum response at short time is enhanced by free energy dissipation. At the same time, the low frequency fluctuations and the adaptation error decrease with the free energy dissipation algebraically and exponentially, respectively. [1] ``Free Energy Cost of Reducing Noise while Maintaining a High Sensitivity'', Pablo Sartori and Yuhai Tu, Phys. Rev. Lett. 2015. 115: 118102.   

Mimicking Nonequilibrium Steady States with Time-Periodic Driving

Under static conditions, a system satisfying detailed balance generically relaxes to an equilibrium state in which there are no currents: to generate persistent currents, either detailed balance must be broken or the system must be driven in a time-dependent manner. A stationary system that violates detailed balance evolves to a nonequilibrium steady state (NESS) characterized by fixed currents. Conversely, a system that satisfies instantaneous detailed balance but is driven by the time-periodic variation of external parameters - also known as a stochastic pump (SP) - reaches a periodic state with non-vanishing currents. In both cases, these currents are maintained at the cost of entropy production. Are these two paradigmatic scenarios effectively equivalent? For discrete-state systems we establish a mapping between NESS and SP. Given a NESS characterized by a particular set of stationary probabilities, currents and entropy production rates, we show how to construct a SP with exactly the same (time-averaged) values. The mapping works in the opposite direction as well. These results establish a proof of principle: they show that SP are able to mimic the behavior of NESS, and vice-versa, within the theoretical framework of discrete-state stochastic thermodynamics.   

The thermal vacuum for non-equilibrium steady state

Our purpose is to construct a theoretical description of non-equilibrium steady state (NESS), employing thermo field dynamics (TFD). TFD is the operator-based formalism of thermal quautum field theory, where every degree of freedom is doubled and thermal averages are given by expectation values of the thermal vacuum\footnote{H. Umezawa, \textit{Thermo Field Dynamics and Condensed States} (Elsevier Science Ltd, 1982).}. To specify the thermal vacuum for NESS is a non-trivial issue, and we attempt it on the analogy between the superoperator formalism and TFD\footnote{M. Schmutz, Z. Phys. B \textbf{30}, 97 (1978); Y. Nakamura and Y. Yamanaka, Ann. Phys. (N.Y.) \textbf{331}, 51 (2013).}. Using the thermal vacuum thus obtained, we analyze the NESS which is realized in the two-reservoir model. It will be shown that the NESS vacuum of the model coincides with the fixed point solutions of the quantum transport equation derived by the self-consistent renormalization of the self-energy in non-equilibrium TFD \footnote{H. Umezawa, \textit{Advanced Field Theory: Micro, Macro, and Thermal Physics} (AIP, 1993); Y. Nakamura and Y. Yamanaka, Ann. Phys. (N.Y.) \textbf{331}, 51 (2013).} .   

Closed hierarchies and non-equilibrium steady states of driven systems

We present a class of tractable non-equilibrium dynamical quantum systems which includes combinations of injection, detection and extraction of particles interspersed by unitary evolution. We show how such operations generate a hierarchy of equations tying lower correlation functions with higher order ones. The hierarchy closes for particular choices of measurements and leads to a rich class of evolutions whose long time behavior can be simulated efficiently. In particular, we use the method to describe the dynamics of current generation through a generalized quantum exclusion process, and exhibit an explicit formula for the long time energy distribution in the limit of weak driving.   

Mechanical autonomous stochastic heat engines

Stochastic heat engines extract work from the Brownian motion of a set of particles out of equilibrium. So far, experimental demonstrations of stochastic heat engines have required extreme operating conditions or nonautonomous external control systems. In this talk, we will present a simple, purely classical, autonomous stochastic heat engine that uses the well-known tension induced nonlinearity in a string. Our engine operates between two heat baths out of equilibrium, and transfers energy from the hot bath to a work reservoir. This energy transfer occurs even if the work reservoir is at a higher temperature than the hot reservoir. The talk will cover a theoretical investigation and experimental results on a macroscopic setup subject to external noise excitations. This system presents an opportunity for the study of non equilibrium thermodynamics and is an interesting candidate for innovative energy conversion devices.   

A dissipation bound for thermodynamic control

Biological and engineered systems operate by coupling function to the transfer of heat and/or particles down a thermal or chemical gradient. In idealized \textit{deterministically} driven systems, thermodynamic control can be exerted reversibly, with no entropy production, as long as the rate of the protocol is made slow compared to the equilibration time of the system. Here we consider \textit{fully realizable, entropically driven} systems where the control parameters themselves obey rules that are reversible and that acquire directionality in time solely through dissipation. We show that when such a system moves in a directed way through thermodynamic space, it must produce entropy that is on average larger than its generalized displacement as measured by the Fisher information metric. This distance measure is sub-extensive but cannot be made small by slowing the rate of the protocol.   

Thermodynamic second law in a feedback process with time delay

We investigate a realistic feedback process repeated in multiple steps where a feedback protocol from measurement is applied with delay and maintains for a finite duration until next step. Unlike a feedback without delay, a composite system consists of the system and two memories where previous and present measurement outcomes are stored, leading to the 3-state Shannon entropy for the composite system. Then according to the thermodynamic second law, the change of the 3-state Shannon entropy provides the upper bound for heat flow from reservoir to system during the feedback and relaxation process. However, if the feedback protocol is depending on memory states sequentially, it turns out that the tighter bound for heat production can be obtained by integrating out the irrelevant memory state. We exemplify a cold damping case where a velocity of a particle is measured and a dissipative protocol is applied by feedback, and it is confirmed that the Shannon-entropy change of the reduced composite system gives the tighter bound for heat production.   

Direct measurement of the Einstein relation in a macroscopic, non-equilibrium system of chaotic surface waves

Equilibrium statistical mechanics is traditionally limited to thermal systems. Can it be applied to athermal, non-equilibrium systems that nonetheless satisfy the basic criteria of steady-state chaos and isotropy? We answer this question using a macroscopic system of chaotic surface waves which is, by all measures, non-equilibrium. The waves are generated in a dish of water that is vertically oscillated above a critical amplitude. We have constructed a rheometer that actively measures the drag imparted by the waves on a buoyant particle, a quantity entirely divorced in origin from the drag imparted by the fluid in which the particle floats. We also perform a separate, passive measurement, extracting a diffusion constant and effective temperature. Having directly measured all three properties (temperature, diffusion constant, and drag coefficient) we go on to show that our macroscopic, non-equilibrium case is wholly consistent with the Einstein relation, a classic result for equilibrium thermal systems.   

Least action and entropy considerations of self-organization in Benard cells

We study self-organization in complex systems using first principles in physics. Our approach involves the principle of least action and the second law of thermodynamics. In far from equilibrium systems, energy gradients cause internal ordering to facilitate the dissipation of energy in the environment. This internal ordering decreases their internal entropy in order to obey the principle of least action, minimizing the product of time and energy for transport through the system. We are considering the connection between action and entropy decrease inside Benard cells in order to derive some general features of self-organization. We are developing mathematical treatment of this coupling and comparing it to results from experiments and simulations.   

Dissipation and irreversibility for models of mechanochemical machines

For biological systems to maintain order and achieve directed progress, they must overcome fluctuations so that reactions and processes proceed forwards more than they go in reverse. It is well known that some free energy dissipation is required to achieve irreversible forward progress, but the quantitative relationship between irreversibility and free energy dissipation is not well understood. Previous studies focused on either abstract calculations or detailed simulations that are difficult to generalize. We present results for mechanochemical models of molecular machines, exploring a range of model characteristics and behaviours. Our results describe how irreversibility and dissipation trade off in various situations, and how this trade-off can depend on details of the model. The irreversibility-dissipation trade-off points towards general principles of microscopic machine operation or process design. Our analysis identifies system parameters which can be controlled to bring performance to the Pareto frontier.