Session A9: Systems Far from Equilibrium I

Directed motion and useful work from an isotropic non-equilibrium distribution

Since the Maxwell demon thought experiment, the extraction of useful work and directed motion from unbiased non-equilibrium distributions has been the source of fascination, intrigue, and confusion. Being a fundamental scientific problem, it is also of signficant practical interest for various biological and nanotechnological applications. We propose a new type of ``motor'' driven by the heat flow between non-equilibrium velocity and equilibrium coordinate distributions. Namely, we demonstrate that a gas of classical particles trapped in an external asymmetric potential undergoes a quasiperiodic motion, if the temperature of its initial velocity distribution Tne differs from the equilibrium temperature Teq. The magnitude of the effect is determined by the value of Tne - Teq, and the direction of the motion is determined by the sign of this expression. The ``loading'' and ``unloading'' of the gas particles change directions of their motion, thereby creating a possibility of shuttle-like motion. The system works as a Carnot engine where the heat flow between kinetic and potential parts of the non-equilibrium distribution produces the useful work. Phys.Rev. E 77, 011115 (2008)   

Coherent transport and heat, entropy fluctuations in a thermal Brownian motor

We investigate the heat, entropy and work fluctuations in a thermal Brownian motor driven by spatially inhomogeneous temperature. We show that the total heat, entropy production and the work fluctuations satisfy the fluctuation therorem in the steady state over finite time trajectories. The transport coherence of the motor, as determined by the Peclet number is also investigated as a function of various parameters of the system.   

Exact results for currents in nonadiabatic stochastic pumps

Biological systems abound with examples of molecular machines: assemblies of molecules that perform specific useful mechanical tasks, such as the motor proteins kinesin and myosin. Remarkably, the first steps in developing useful artificial molecular motors have been taken with the synthesis and manipulation of molecular complexes such as catenanes and rotaxanes. These developments have spurred an interest in developing theoretical frameworks which describe these mesoscopic machines that operate in the presence of thermal noise. In this talk I will analyze a generic model of molecular machines known as stochastic pumps in which useful directed motion (or current) is produced by the variation of external parameters. The main result is an exact expression for the current in the presence of nonadiabatic pumping. This expression connects to a variety of results from the field of brownian ratchets and leads to a surprising ``no-pumping'' theorem: a set of conditions that guarantee no excess or pumped current. These predictions also agree with the observations on catenanes, interlocked ring molecules, made by Leigh et. al. [Nature, 424, 174 (2003)].   

Rapidly forced quantum Brownian motion

We study the steady state behaviour of a confined quantum Brownian particle which is subjected to a space-dependent, rapidly oscillating time-periodic force. To leading order in the period of driving, the result of the oscillating force is to produce an effective static potential which has a quantum contribution V$_{q}$ which adds on to the classical result. This is shown by using a coherent state representation of bath oscillators which leads to a c-number generalized quantum Langevin equation. We evaluate V$_{q}$ exactly in the case of an Ohmic dissipation bath and show that it takes on different forms in different regimes, determined by the ratio of the thermal wavelength to the spatial spread of the driving force.   

Deciding how far is far from equilibrium

Nonequilibrium systems have both fundamental and technological interest for their unusual behavior with research efforts mainly focused on their properties. Surprisingly, little research effort has been put on diagnosing how far a system is from equilibrium. Clearly, addressing such an issue is of fundamental and technological relevance. To this end, we present results of how a particle in contact with a heat reservoir is being driven away from equilibrium by a time dependent potential well. The methodology can be straightforwardly extended to systems with more particles and under the influence of more realistic potentials.   

Random Sequential Adsorption on patterned substrates: jammed state structure and kinetic properties

The irreversible adsorption on a patterned substrate is studied through extensive Monte Carlo simulations. As a pattern, we adopted square cells positioned at the vertices of a square lattice. Particles attempting adsorption can only stick to the substrate if they do not overlap previously adsorbed ones (excluded volume interaction) and if their geometrical centers land inside a cell. Once a particle is adsorbed, it does not detach from or diffuse on the substrate, thus representing an extended random sequential adsorption model. The distribution of particles sizes follows a truncated gaussian-size distribution with values of the size dispersion varying from zero (monodisperse) to $20\%$ (polydisperse) of the mean particle radius. We address the influence of both the pattern and size dispersion on the jammed state structure. We also present results on how the kinetics of approach to the jammed state is affected by the particular values taken by parameters like cell size and cell-cell separation and show that they can lead to either exponential or power-law functional dependences.   

Phase Transition with Non-Thermodynamic States in Reversible Polymerization

We investigate a reversible polymerization process in which individual polymers aggregate and fragment at a rate proportional to their molecular weight. We find a nonequilibrium phase transition despite the fact that the dynamics are perfectly reversible. When the strength of the fragmentation process exceeds a critical threshold, the system reaches a thermodynamic steady state where the total number of polymers is proportional to the system size. The polymer length distribution has a sharp exponential tail in this case. When the strength of the fragmentation process falls below the critical threshold, the steady state becomes non-thermodynamic as the total number of polymers grows sub-linearly with the system size. Moreover, the length distribution has an algebraic tail and the characteristic exponent varies continuously with the fragmentation rate.   

Stochastic continuum theory of active nematics

We derive a stochastic continuum theory of active nematics by direct coarse-graining of a generic microscopic model and study it numerically. This allows to clarify the microscopic origin of the various terms found and to determine the non-trivial structure of the noises. We show in particular that two terms coupling density and order the non-equilibrium active current argued before to be at the origin of giant density fluctuations, and a multiplicative conserved noise are necessary to obtain a faithful description of the original model.   

Fokker-Planck Dynamics in the Energy Domain

We derive a Fokker-Planck Equation (FPE) in the energy domain for a system in an infinite heat bath by coarse-graining its microscopic Master Equation. The resulting FPE carries information on the dynamics through a function $\lambda(E)$, which is a sum over all possible transitions given a state of energy E. We investigate the effects of changing the assumptions about the transition rates without changing the Hamiltonian of the model. By determining the eigenvalues of the equivalent Schrodinger Equation (SE), we get the relaxation spectrum of the FPE. We find that in the thermodynamic limit the equivalent SE approaches the classical limit, and we use the WKB approximation to solve it. We illustrate the use of the method by applying it to several examples, including a system of harmonic oscillators, and a paramagnet in an external magnetic field.   

Doing the Impossible: Very Rare Events in the Harmonic Measure

We have developed a method of obtaining accurate data of rare events using biased sampling of random walkers. We have obtained the harmonic measure, analogous to the perpendicular electric field on a charged conductor, for percolation, Ising model, and Diffusion Limited Aggregation (DLA) clusters. We measured probabilities down to 10$^{-300}$ for percolation and Ising model clusters. These small probabilities allowed us to verify the theoretical predictions for the harmonic measure made by Duplantier. For DLA, which has no theory, we obtained probabilities down to 10$^{-100}$. The previous lowest probability was obtained using iterative conformal maps and was limited to small clusters and comparatively high probabilities. For all systems we have obtained the generalized dimension Dq, the singularity spectrum f(alpha), and the distribution of probabilities.   

Anisotropic 2-dimensional Robin Hood model

We have considered the Robin Hood model introduced by Zaitsev[1] to discuss flux creep and depinning of interfaces in a two dimensional system. Although the model has been studied extensively analytically in 1-d [2], its scaling laws have been verified numerically only in that case. Recent work suggest that its properties might be important to understand surface friction[3], where its 2-dimensional properties are important. We show that in the 2-dimensional case scaling laws can be found provided one considers carefully the anisotropy of the model, and different ways of introducing that anisotropy lead to different exponents and scaling laws, in analogy with directed percolation, with which this model is closely related[4]. We show that breaking the rotational symmetry between the \textbf{x} and \textbf{y} axes does not change the scaling properties of the model, but the introduction of a preferential direction of accretion (``robbing'' in the language of the model) leads to new scaling exponents. [1] S.I.Zaitsev, Physica \textbf{A189}, 411 (1992) [2] M. Pacuzki, S. Maslov and P.Bak, Phys Rev. \textbf{E53}, 414 (1996) [3] S. Buldyrev, J. Ferrante and F. Zypman Phys. Rev \textbf{E64}, 066110 (2006) [4] G. Odor, Rev. Mod. Phys. \textbf{76}, 663 (2004) .   

A Cellular Automaton Model of Catastrophic Failure

We introduce a two-dimensional cellular automaton model for studying the catastrophic failure of materials under stress. Our model is similar to the Olami-Feder-Christensen earthquake model [Z.\ Olami \emph{et al.}, Phys.\ Rev.\ Lett.\ \textbf{68}, 1244 (1992)] except that after a site fails $f$-times, it no longer can receive stress from its neighbors. In the limit that the interaction range, $R$, goes to infinity, our model is equivalent to the global load sharing fiber bundle model of Pierce [F.\ T.Pierce, J.\ Text.\ Ind.\ \textbf{17}, 355 (1926)] and Daniels [H.\ E.\ Daniels, Proc.\ Roy.\ Soc.\ London A \textbf{183}, 405 (1945)]. By varying the interaction range, we observe two qualitatively different failure modes. For $R \gg 1$ catastrophic failure resembles a nucleation-like event which grows symmetrically from a single initiating site and fails every site in the lattice. In contrast, for $R \approx 1$ a percolating cluster of failed sites spans the system despite the many active sites that persist, even after catastrophic failure. We use the stress-fluctuation metric to study the ergodicity of our model and hence the validity of equilibrium descriptions of fracture.   

Heat transport in quantum spin chains: the relevance of integrability

Heat transport in quantum spin chains is investigated through the master equation in Lindblad form derived from the Schroedinger equation of a system coupled with two baths via the projector operator technique. We find that the Fourier's Law of heat transport is obeyed in some systems. Although a general proof has not been established, after a survey of various quantum spin chains, our results suggest the criteria of anomalous heat transport is not the integrability of the Hamiltonian, but whether or not it can be mapped to non- interacting fermions.   

Hard-core Bosons in time-varying traps

We present a study of the time evolution of hard-core bosons (HCBs) in a one-dimensional, time-varying optical trap. Previous results have shown that one-dimensional HCBs can form superfluid and Mott-insulator phases. Using an exact numerical approach, we study the dynamics of the system when the trap curvature is modulated. We find the dynamics to be markedly different in the two phases, and address its relevance in the observation of these phases in optical lattice experiments.   

Control of Transport Behavior in spin-1/2 Heisenberg Systems

A complete understanding of transport behavior in many-body systems is one of the utmost challenges in fundamental studies of nonequilibrium statistical mechanics. In the classical domain, it is widely believed that chaotic systems should show diffusive transport, whereas integrability should be associated with ballistic transport. In the quantum domain, the conditions that determine specific transport behaviors are still under debate. Here, we analyze transport of local magnetization in finite spin-1/2 Heisenberg systems. By adjusting parameters in the Hamiltonian, these quantum systems may show both integrable and chaotic limits. We provide examples of chaotic systems leading to diffusive and also to ballistic transport. In addition, we develop schemes to control the transport behavior in these systems, showing that quantum control methods may be used to induce a transition from diffusive to ballistic transport.