Session B29: Fluctuations in Non-Equilibrium Systems

Large rare fluctuations in systems with delayed dissipation

We study the probability distribution and the escape rate in noise-driven nonlinear systems with delayed dissipation. Accounting for the delay requires a significant modification of the conventional rare events theory. We develop the corresponding general formulation and find explicit results in the limiting cases. To logarithmic accuracy in the fluctuation intensity, the problem is reduced to a variational problem. It describes the most probable path followed by the system in the random rare event of interest. In contrast to Markov systems, the equations for the most probable paths are acausal due to the delay. If the dissipation and noise come from the coupling to a thermal bath, they are related by the fluctuation-dissipation relation, but our results are not limited to this case. In thermal equilibrium, the most probable path passing through a remote state has time reversal symmetry. However, again in contrast to Markov systems, one cannot uniquely define a path that starts from a state with given system coordinate and momentum. The corrections to the logarithm of the probability distribution and the escape activation energy for small dissipation delay and small noise correlation time are obtained in explicit form.   

Thermal rectification in non-linear structures with bulk losses

A mechanism for thermal rectification based on the interplay between non-uniform bulk losses with nonlinearity is presented. We theoretically analyze the phenomenon using an anharmonic array of coupled oscillators coupled to the left and right with two Langevin reservoirs. A third probe thermostat (with temperature $T_B$) is placed in an asymmetric position in the bulk of the lattice thus breaking the translational symmetry and leading to rectification of heat flow. We note that for $T_B=0$ this Langevin term is equivalent to a simple friction. We find that an increase of the friction strength can increase both the asymmetry and heat flux.   

An exactly solvable model of Maxwell's demon

The paradox of Maxwell's demon has stimulated numerous thought experiments, leading to discussions about the thermodynamic implications of information processing. However, the field has lacked a tangible example or model of an autonomous, mechanical system that reproduces the actions of the demon. To address this issue, we introduce an explicit model of a device that can deliver work to lift a mass against gravity by rectifying thermal fluctuations, while writing information to a memory register. We solve for the steady-state behavior of the model and construct its nonequilibrium phase diagram. In addition to the engine-like action described above, we identify a ``Landauer eraser'' region in the phase diagram where the model uses externally supplied work to remove information from the memory register. Our model offers a simple paradigm for investigating the thermodynamics of information processing by exposing a transparent mechanism of operation.   

The thermodynamics of prediction

We expose the fundamental equivalence between model inefficiency and thermodynamic inefficiency, measured by dissipation. The dynamics of any system responding to a stochastic environmental signal can be interpreted as computing an implicit model of the driving signal. The system's state retains information about past environmental fluctuations, and a fraction of this information is predictive of future fluctuations. The remaining nonpredictive information reflects model complexity that does not improve predictive power, and thus represents the inefficiency of the model. We find that instantaneous nonpredictive information: 1) is proportional to the work dissipated due to environmental change; 2) provides a lower bound on the total average dissipated work when summed over the length of a driving protocol; 3) augments the lower bound on heat generated due to information erasure (Landauer's principle). Our results hold far from thermodynamic equilibrium and are thus applicable to a wide range of systems, including biomolecular machines. They highlight a profound connection between the effective use of information and efficient thermodynamic operation: any system constructed to keep memory about its environment and to operate with maximal energetic efficiency has to be predictive.   

Geometry of thermodynamic control

A fundamental problem in modern thermodynamics is how a molecular-scale machine performs useful work, while operating away from thermal equilibrium without excessive dissipation. We show that when a thermodynamic system is driven from equilibrium, in the linear response regime, the space of controllable parameters has a Riemannian geometry induced by a generalized friction tensor. This metric structure controls the dissipation of finite-time transformations, and bestows optimal protocols (geodesics on the Riemannian manifold) with many useful properties. We exploit this geometric insight to construct closed-form expressions for minimal-dissipation protocols for a model system of a particle diffusing in a one-dimensional harmonic potential, where the spring constant, inverse temperature, and trap location are adjusted simultaneously. This simple model has a surprisingly rich geometry, which we test via a numerical implementation of the Fokker-Planck equation.   

Clinical application of fluctuation dissipation theory - Prediction of heart rate response to spontaneous breathing trial

Contrary to the traditional view of the healthy physiological state as being a single static state, variation in physiologic variables has more recently been suggested to be a key component of the healthy state. Indeed, aging and disease are characterized by a loss of such variability. We apply the conceptual framework of fluctuation-dissipation theory (FDT) to predict the response to a common clinical intervention from historical fluctuations in physiologic time series data. The non-equilibrium FDT relates the response of a system to a perturbation to natural fluctuations in the stationary state of the system. We seek to understand with the FDT a common clinical perturbation, the spontaneous breathing trial (SBT), in which mechanical ventilation is briefly suspended while the patient breathes freely for a period of time. As a stress upon the heart of the patient, the SBT can be characterized as a perturbation of heart rate dynamics. A non-equilibrium, but steady-state FDT allows us to predict the heart rate recovery after the SBT stress. We show that the responses of groups of similar patients to the spontaneous breathing trial can be predicted by this approach. This mathematical framework may serve as part of the basis for personalized critical care.   

Cumulant generating function formula of heat transfer in ballistic systems with lead-lead coupling and general nonlinear systems

Based on a two-time observation protocol, we consider heat transfer in a given time interval $t_M$ in a lead-junction-lead system taking coupling between the leads into account. In view of the two-time observation, consistency conditions are carefully verified in our specific family of quantum histories. Furthermore, its implication is briefly explored. Then using the nonequilibrium Green's function method, we obtain an exact formula for the cumulant generating function for heat transfer between the two leads, valid in both transient and steady-state regimes. Also, a compact formula for the cumulant generating function in the long-time limit is derived, for which the Gallavotti-Cohen fluctuation symmetry is explicitly verified. In addition, we briefly discuss Di Ventra's repartitioning trick regarding whether the repartitioning procedure of the total Hamiltonian affects the nonequilibrium steady-state current fluctuation. All kinds of properties of nonequilibrium current fluctuations, such as the fluctuation theorem in different time regimes, could be readily given according to these exact formulas. Finally a practical formalism dealing with cumulants of heat transfer across general nonlinear quantum systems is established based on field theoretical/algebraic method.   

Fluctuation Relations for Current Components in Mesoscopic Electric Circuits

Discovery of Fluctuation Theorems (FTs) for non-equilibrium systems led to optimism that they might serve as universal laws that had long been missing from the study of nonequilibrium systems. Surprisingly, recent experimental work has shown that the FTs can fail in an electric circuit, but could be salvaged under the experimental conditions if the affinity parameter is suitably renormalized by a factor of 0.1. Motivated by this new experimental result we present a new class of fluctuation relations, to which we will refer as ``Fluctuation Relations for Current Components'' (FRCCs). Unlike standard fluctuation theorems, FRCCs follow from the seemingly trivial fact that to know statistics of particle currents, it is sufficient to know only statistics of single particle geometric trajectories while the information about time moments, at which particles make transitions along such trajectories, is irrelevant. We also show that FRCCs are robust in the sense that they do not depend on some basic types of electron interactions and some quantum coherence effects.   

The Dependence of Heat Fluctuation Theorem on an Initial Distribution

The fluctuation theorem (FT) proven for work does not hold for heat even in the long time limit. As the two quantities differ by the change in energy at the initial and final times, we suspect that the memory of an initial distribution may remain in the heat production accumulated for a long time. We investigate the dependence of the large deviation function (LDF) and FT on the temperature of the initial equilibrium distribution for the motion of a Brownian particle in a harmonic potential dragged with a constant velocity. The conventional saddle point integration for the LDF used in van Zon and Cohen, Phys.\ Rev.\ Lett.\ {\bf 91}, 110601 (2003) is found to fail as the saddle point approaches asymptotically the singularity at the branch point in the long time limit. We develop a new mathematical method to resolve this problem and confirm it with numerical simulations. As a result, the tail of LDF, i.e., a region of rare events, is shown to depend remarkably on the initial temperature and also causes more types of modifications of FT's than the so called extended FT proposed by van Zon and Cohen. We expect that our method can be applied to the investigation of the dependence of initial memories in other nonequilibrium systems.   

Threshold for everlasting initial memory for rare events in equilibration processes

Conventional wisdom indicates that initial memory should decay away exponentially in time for general (noncritial) equilibration processes. In particular, time-integrated quantities such as heat are presumed to lose initial memory in a sufficiently long-time limit. However, we show that the large deviation function of time-integrated quantities may exhibit initial memory effect even in the infinite-time limit, if the system is initially prepared sufficiently far away from equilibrium. For a Brownian particle dynamics, as an example, we found a sharp finite threshold rigorously, beyond which the corresponding large deviation function contains everlasting initial memory. The physical origin for this phenomenon is explored with an intuitive argument and also from a toy model analysis.   

Fluctuation theorems and entropy production with odd-parity variables

We show that the total entropy production in stochastic processes with odd-parity variables (under time reversal) is separated into three parts, only two of which satisfy the integral fluctuation theorems in general. One is the usual excess contribution, which can appear only transiently and is called non-adiabatic. Another one is attributed solely to the breakage of detailed balance. The last part not satisfying the fluctuation theorem comes from the steady-state distribution asymmetry for odd-parity variables, which is activated in a non-transient manner. The latter two parts combine together as the house-keeping (adiabatic) contribution, whose positivity is not guaranteed except when the excess contribution completely vanishes. Our finding reveals that the equilibrium requires the steady-state distribution symmetry for odd-parity variables independently, in addition to the usual detailed balance.   

A novel nature in nonequilibrium entropy production with odd-parity variables

We present our recent finding about a novel nature in nonequilibrium entropy production for systems with odd-parity variables under time reversal. In the presence of irreversible forces the entropy production $\Delta S_{env}$ transferred from system to environment is not equal to $Q/T$ where $Q$ is the heat transfer and $T$ the temperature of heat bath. We consider a dissipative force applied by external agent in addition to that given by heat bath. Then $\Delta S_{env}$ has extra contribution to $Q/T$ for which an appropriate physical explanation is still open. Another example for irreversible force is a form of $-A\cdot\vec{p}/m$ for antisymmetric matrix $A$ which is realized by a Lorentz force in a uniform magnetic field. In spite of no heat dissipation $\Delta S_{env}$ has a nonvanishing positive contribution. We find that it is due to a nonzero phase space current remaining through stochastic average, which is in fact a nonzero average force. Basically it plays the same role as a nonzero position space current observed in system with even variables only. We suppose interesting situations for different types of irreversible forces.   

About the Equivalence of Phase Retrieval Methods Employed in Nonlinear Spectroscopy and Microscopy

It is well known that the generalized Kramers-Kronig relationship is able to retrieve the phase of a signal from measured power spectra. This phase recovery is a critical procedure in nonlinear optical spectroscopy, e.g. coherent Raman time domain or frequency domain spectroscopy. Several other methods have been developed and being used in the past: notably, nonlinear fitting and maximum entropy method. A firm mathematical comparison of the methods including the effects of final signal sampling and their merit of fidelity will be presented. Attention is given to numerical implementation of the phase retrieval procedure to put it into practice in coherent anti-Stokes Raman microscopy. Phase retrieval examples using all the above methods are taken from earlier and recently recorded spectra.   

Theory of zero-bias anomaly in low-temperature inelastic tunneling spectroscopy

A small zero-bias anomaly (ZBA) in inelastic tunneling spectroscopy (IETS) through nonmagnetic quantum wires has been suggested experimentally at low temperatures [1,2]. Here, the mechanism is discussed theoretically with special attention paid to contributions from low energy phonons [3]. Our theoretical calculations, using an electron-phonon coupling model, predict the ZBA. While experimental information is still limited, our theoretical result agrees with existing experiments. The theory provides useful information, characterizing the ZBA in a nonmagnetic junction.\\[4pt] [1] L. F. Spietz, Ph.D. dissertation, Yale University, 2006.\\[0pt] [2] Y. Selzer, M. A. Cabassi, T. S. Mayer, and D. L. Allara, Nanotechnology 15, S483 (2004).\\[0pt] [3] Y. Asai, Phys. Rev. B Rapid Commun., in press.   

Computation of Microcanonical Entropy Differences in Atomistic Computer Simulation

In this work, two alternative methods to compute thermodynamic entropy differences $\Delta S=S(E_2)-S(E_1)$ between two microcanonical states (produced via atomistic computer simulation, either deterministic or stochastic) at total energies $E_1$ y $E_2$ are presented. The first method is straightforward to implement, as it only needs potential energy samples from both simulations; however, it requires that fluctuations of potential energy are similar in magnitude to the energy difference $\Delta E$ between the states. It is therefore best suited for simulations in small systems (hundred of atoms). The second method, based on Bayesian probability and information theory, removes this limitation: it allows the computation of the entropy curve $S(E)$ for a wide range of energies and therefore is a viable alternative to methods such as Wang-Landau Monte Carlo. It is based on inferring the configurational density of states (CDOS) from potential energy samples. A simple model for the CDOS of embedded atom metals is presented and tested in Au and Cu by computing entropy and free energy differences.