State Of Maximum Entropy Research Articles

Self-Energy Closure for Inhomogeneous Turbulent Flows and Subgrid Modeling

A new statistical dynamical closure theory for general inhomogeneous turbulent flows and subgrid modeling is presented. This Self-Energy (SE) closure represents all eddy interactions through nonlinear dissipation or forcing ‘self-energy’ terms in the mean-field, covariance and response function equations. This makes the renormalization of the bare dissipation and forcing, and the subgrid modeling problem, transparent. The SE closure generalizes the quasi-diagonal direct interaction closure to allow for more complex interactions. The SE closure is applicable to flows in different geometries, is exact near maximum entropy states corresponding to canonical equilibrium, and provides a framework for deriving simpler realizable closures.

Initial states and infrared physics in locally de Sitter spacetime

The long wavelength physics in a de Sitter region depends on the initial quantum state. While such long wavelength physics is under control for massive fields near the Hartle-Hawking vacuum state, such initial states make unnatural assumptions about initial data outside the region of causal contact of a local observer. We argue that a reasonable approximation to a maximum entropy state, one that makes minimal assumptions outside an observer's horizon volume, is one where a cutoff is placed on a surface bounded by timelike geodesics, just outside the horizon. For sufficiently early times, such a cutoff induces secular logarithmic divergences with the expansion of the region. For massive fields, these effects sum to finite corrections at sufficiently late times. The difference between the cutoff correlators and Hartle-Hawking correlators provides a measure of the theoretical uncertainty due to lack of knowledge of the initial state in causally disconnected regions. These differences are negligible for primordial inflation, but can become significant during epochs with very long-lived de Sitter regions, such as we may be entering now.

Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas

The problem of how complex quantum systems eventually come to rest lies at the heart of statistical mechanics. The maximum entropy principle put forward in 1957 by E. T. Jaynes suggests what quantum states one should expect in equilibrium but does not hint as to how closed quantum many-body systems dynamically equilibrate. A number of theoretical and numerical studies accumulate evidence that under specific conditions quantum many-body models can relax to a situation that locally or with respect to certain observables appears as if the entire system had relaxed to a maximum entropy state. In this work, we report the experimental observation of the non-equilibrium dynamics of a density wave of ultracold bosonic atoms in an optical lattice in the regime of strong correlations. Using an optical superlattice, we are able to prepare the system in a well-known initial state with high fidelity. We then follow the dynamical evolution of the system in terms of quasi-local densities, currents, and coherences. Numerical studies based on the time-dependent density-matrix renormalization group method are in an excellent quantitative agreement with the experimental data. For very long times, all three local observables show a fast relaxation to equilibrium values compatible with those expected for a global maximum entropy state. We find this relaxation of the quasi-local densities and currents to initially follow a power-law with an exponent being significantly larger than for free or hardcore bosons. For intermediate times the system fulfills the promise of being a dynamical quantum simulator, in that the controlled dynamics runs for longer times than present classical algorithms based on matrix product states can efficiently keep track of.

Thermodynamic Analysis of Biological Systems

Copyright: © 2012 Ram D. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. The laws of thermodynamics traditionally describe the physical quantities (energy, entropy, temperature, volume, etc) of a thermodynamic system and the relationship between heat and work. Given that these laws are universal and represent the epitome of physical sciences, it is somewhat disconcerting to note that biologists have not always had a comfortable relationship with thermodynamics! This discomfort originates in the seemingly irreconcilable differences between the physical (simple) processes described by thermodynamics and the biological (complex) processes. As we all know, thermodynamic system tend to equilibrium, which for an isolated system implies a state of maximum disorder or maximum entropy. However, almost all biological systems do not ever reach equilibrium and seem to “evolve” to states of higher order, rather than disorder. Clearly, this discrepancy can be explained by simply noting that biological (living) systems are not isolated and are not in equilibrium with their surroundings. But this only explains why biological systems can be ordered but it does not explain how they came to be so. Of all the innumerable processes that increase entropy in the universe why does nature choose to create order in a few of its subsystems? Simply, put why does life create order out of disorder?

Kadanoff-Baym Approach to Entropy Production in O(N) Theory with Next-to-Leading Order Self-Energy

Abstract We investigate entropy production in the O(N) scalar theory using the Kadanoff-Baym equation. We show that one of the candidate expressions of the kinetic entropy satisfies the H-theorem in the first order of the gradient expansion with the next-to-leading-order self-energy of the 1/N expansion in the symmetric phase, and that entropy production occurs as the Green's function evolves with nonzero collision term contributions. Entropy production stops at local thermal equilibrium where the collision term contribution vanishes and the maximal entropy state is realized. We numerically examine these features of entropy production in thermalization processes in 1+1 dimensions for a couple of homogeneous cases, where the thermalization can proceed only with the off-shell effects. We find that the entropy production rate γ is larger for smaller N and is found to follow γ ∝ (1/N)ν where δ ≳ 2 at strong coupling measured in the unit of bare mass (m), ⋋= 40 m2.

Shape memory polymers for active cell culture.

Shape memory polymers (SMPs) are a class of "smart" materials that have the ability to change from a fixed, temporary shape to a pre-determined permanent shape upon the application of a stimulus such as heat(1-5). In a typical shape memory cycle, the SMP is first deformed at an elevated temperature that is higher than its transition temperature, T(trans;) [either the melting temperature (T(m;)) or the glass transition temperature (T(g;))]. The deformation is elastic in nature and mainly leads to a reduction in conformational entropy of the constituent network chains (following the rubber elasticity theory). The deformed SMP is then cooled to a temperature below its T(trans;) while maintaining the external strain or stress constant. During cooling, the material transitions to a more rigid state (semi-crystalline or glassy), which kinetically traps or "freezes" the material in this low-entropy state leading to macroscopic shape fixing. Shape recovery is triggered by continuously heating the material through T(trans;) under a stress-free (unconstrained) condition. By allowing the network chains (with regained mobility) to relax to their thermodynamically favored, maximal-entropy state, the material changes from the temporary shape to the permanent shape. Cells are capable of surveying the mechanical properties of their surrounding environment(6). The mechanisms through which mechanical interactions between cells and their physical environment control cell behavior are areas of active research. Substrates of defined topography have emerged as powerful tools in the investigation of these mechanisms. Mesoscale, microscale, and nanoscale patterns of substrate topography have been shown to direct cell alignment, cell adhesion, and cell traction forces(7-14). These findings have underscored the potential for substrate topography to control and assay the mechanical interactions between cells and their physical environment during cell culture, but the substrates used to date have generally been passive and could not be programmed to change significantly during culture. This physical stasis has limited the potential of topographic substrates to control cells in culture. Here, active cell culture (ACC) SMP substrates are introduced that employ surface shape memory to provide programmed control of substrate topography and deformation. These substrates demonstrate the ability to transition from a temporary grooved topography to a second, nearly flat memorized topography. This change in topography can be used to control cell behavior under standard cell culture conditions.

Oseen vortex as a maximum entropy state of a two dimensional fluid

During the last four decades, a considerable number of investigations has been carried out into the evolution of turbulence in two dimensional Navier-Stokes flows. Much of the information has come from numerical solution of the (otherwise insoluble) dynamical equations and thus has necessarily required some kind of boundary conditions: spatially periodic, no-slip, stress-free, or free-slip. The theoretical framework that has proved to be of the most predictive value has been one employing an entropy functional (sometimes called the Boltzmann entropy) whose maximization has been correlated well in several cases with the late-time configurations into which the computed turbulence has relaxed. More recently, flow in the unbounded domain has been addressed by Gallay and Wayne who have shown a late-time relaxation to the classical Oseen vortex (also sometimes called the Lamb-Oseen vortex) for situations involving a finite net circulation or non-zero total integrated vorticity. Their proof involves powerful but difficult mathematics that might be thought to be beyond the preparation of many practicing fluid dynamicists. The purpose of this present paper is to remark that relaxation to the Oseen vortex can also be predicted in the more intuitive framework that has previously proved useful in predicting computational results with boundary conditions: that of an appropriate entropy maximization. The results make no assumption about the size of the Reynolds numbers, as long as they are finite, and the viscosity is treated as finite throughout.

Proof of a Modified Jaynes's Estimation Theory

It is proved that the state of maximum entropy, having observed values for the n observables, X1,…,Xn, is the same state that minimises the matrix of covariances of any n locally unbiased estimators for n parameters for the probability distribution of X1,…,Xn. We sketch how to get a similar result in quantum theory, in which X1,…,Xn are (not necessarily commuting) quadratic forms that are bounded relative to a positive self-adjoint operator H such that exp (-βH) is of trace-class for some positive β.

Quantum dynamics of a four-well Bose-Hubbard model with two different tunneling rates

We consider a theoretical model of a four-mode Bose-Hubbard model consisting of two pairs of wells coupled via two processes with two different rates. The model is naturally divided into two subsystems with strong intra-system coupling and much weaker coupling between the two subsystems and has previously been introduced as a model for Josephson heat oscillations by Strzys and Anglin [\pra {\bf 81}, 043616 (2010) ]. We examine the quantum dynamics of this model for a range of different initial conditions, in terms of both the number distribution among the wells and the quantum statistics. We find that the time evolution is different to that predicted by a mean-field model and that this system exhibits a wide range of interesting behaviours. We find that the system equilibriates to a maximum entropy state and is thus a useful model for quantum thermalisation. As our model may be realised to a good approximation in the laboratory, it becomes a candidate for experimental investigation.

Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states

A simplified thermodynamic approach of the incompressible 2D Euler equation is considered based on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibrium states are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these sole constraints. The vorticity fluctuations are Gaussian while the mean flow is characterized by a linear $\bar{\omega}-\psi$ relationship. Furthermore, the maximization of entropy at fixed energy, circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophy at fixed energy and circulation. This provides a justification of the minimum enstrophy principle from statistical mechanics when only the microscopic enstrophy is conserved among the infinite class of Casimir constraints. A new class of relaxation equations towards the statistical equilibrium state is derived. These equations can provide an effective description of the dynamics towards equilibrium or serve as numerical algorithms to determine maximum entropy or minimum enstrophy states. We use these relaxation equations to study geometry induced phase transitions in rectangular domains. In particular, we illustrate with the relaxation equations the transition between monopoles and dipoles predicted by Chavanis and Sommeria [J. Fluid. Mech. 314, 267 (1996)]. We take into account stable as well as metastable states and show that metastable states are robust and have negative specific heats. This is the first evidence of negative specific heats in that context. We also argue that saddle points of entropy can be long-lived and play a role in the dynamics because the system may not spontaneously generate the perturbations that destabilize them.

Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

Using a Maximum Entropy Production Principle (MEPP), we derive a new type of relaxation equations for two-dimensional turbulent flows in the case where a prior vorticity distribution is prescribed instead of the Casimir constraints [R. Ellis, K. Haven, B. Turkington, Nonlinearity 15, 239 (2002)]. The particular case of a Gaussian prior is specifically treated in connection to minimum enstrophy states and Fofonoff flows. These relaxation equations are compared with other relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776 (1992)] and Chavanis [Physica D 237, 1998 (2008)]. They can serve as numerical algorithms to compute maximum entropy states and minimum enstrophy states with appropriate constraints. We perform numerical simulations of these relaxation equations in order to illustrate geometry induced phase transitions in geophysical flows.

Statistical mechanics of Beltrami flows in axisymmetric geometry: Theory reexamined

A simplified thermodynamic approach of the incompressible axisymmetric Euler equations is considered based on the conservation of helicity, angular momentum, and microscopic energy. Statistical equilibrium states are obtained by maximizing the Boltzmann entropy under these sole constraints. We assume that these constraints are selected by the properties of forcing and dissipation. The fluctuations are found to be Gaussian, while the mean flow is in a Beltrami state. Furthermore, we show that the maximization of entropy at fixed helicity, angular momentum, and microscopic energy is equivalent to the minimization of macroscopic energy at fixed helicity and angular momentum. This provides a justification of this selective decay principle from statistical mechanics. These theoretical predictions are in good agreement with experiments of a von Kármán turbulent flow and provide a way to measure the temperature of turbulence and check fluctuation-dissipation relations. Relaxation equations are derived that could provide an effective description of the dynamics toward the Beltrami state and the progressive emergence of a Gaussian distribution. They can also provide a numerical algorithm to determine maximum entropy states or minimum energy states.

Entropy principle and complementary second law of thermodynamics for self-gravitating systems

(abbreviated) The statistical mechanics of self-gravitating systems is a long-held puzzle. In this work, we employ a phenomenological entropy form of ideal gas, first proposed by White & Narayan, to revisit this issue. By calculating the first-order variation of the entropy, subject to the mass- and energy-conservation constraints, we obtain an entropy stationary equation. Incorporated with the Jeans equation, and by specifying some functional form for the anisotropy parameter beta, we numerically solve the two equations, and demonstrate that the velocity anisotropy parameter plays an important role to attain a density profile that is finite in mass, energy, and spatial extent. If incorporated again with some empirical density profile from simulations, our theoretical predictions of the anisotropy parameter, and the radial pseudo-phase-space density in the outer non-gravitationally degenerate region of the dark matter halo, agree rather well with the simulation data, and the predictions are also acceptable in the middle weak-degenerate region of the dark halo. The second-order variational calculus reveals the seemingly paradoxical but actually complementary consequence that the equilibrium state of self-gravitating systems is the global minimum entropy state for the whole system under long-range violent relaxation, but simultaneously the local maximum entropy state for every and any small part of the system under short-range two-body relaxation and Landau damping. This minimum-maximum entropy duality means that the standard second law of thermodynamics needs to be re-expressed or generalized for self-gravitating systems. We believe that our findings, especially the complementary second law of thermodynamics, may provide crucial clues to the development of the statistical physics of self-gravitating systems as well as other long-range interaction systems.

A quantum central limit theorem for non-equilibrium systems: exact local relaxation of correlated states

We prove that quantum many-body systems on a one-dimensional lattice locally relax to Gaussian states under non-equilibrium dynamics generated by a bosonic quadratic Hamiltonian. This is true for a large class of initial states—pure or mixed—which have to satisfy merely weak conditions concerning the decay of correlations. The considered setting is a proven instance of a situation where dynamically evolving closed quantum systems locally appear as if they had truly relaxed, to maximum entropy states for fixed second moments. This furthers the understanding of relaxation in suddenly quenched quantum many-body systems. The proof features a non-commutative central limit theorem for non-i.i.d. random variables, showing convergence to Gaussian characteristic functions, giving rise to trace-norm closeness. We briefly link our findings to the ideas of typicality and concentration of measure.

Physical foundations of evolutionary theory

The theory of evolution by natural selection is herein subsumed by the 2nd law of thermodynamics. The mathematical form of evolutionary theory is based on a re-examination of the probability concept that underlies statistical physics. Probability regarded as physical must include, in addition to isoenergic combinatorial configurations, also energy in conditional circumstances. Consequently, entropy as an additive logarithmic probability measure is found to be a function of the free energy, and the process toward the maximum entropy state is found equivalent to evolution toward the free energy minimum in accordance with the basic maxim of chemical thermodynamics. The principle of increasing entropy when given as an equation of motion reveals that expansion, proliferation, differentiation, diversification, and catalysis are all ways for a system to evolve toward the stationary state in its respective surroundings. Intriguingly, the equation of evolution cannot be solved when there remain degrees of freedom to consume the free energy, and hence evolutionary trajectories of a nonHamiltonian system remain intractable. Finally, when to-and-from flows of energy are balanced between a system and its surroundings, the system is at the Lyapunov-stable stationary state. The principle of maximal energy dispersal, equivalent to the maximal rate of entropy production, gives rise to the ubiquitous characteristics, conventions, and regularities found in nature, where thermodynamics makes no demarcation line between animate and inanimate.

Maximum entropy condition in multiserver queueing system

Maximum entropy condition in queueing theory of Silviu Guiasu has been subjected to further study in the present investigation. We have derived interarrival time by maximizing the entropy condition using gamma distribution. The paper ends with interesting and useful results in the form of two theorems, where the probability distribution and maximum entropy condition of the possible states of the system for multiserver queueing system has been derived.

Separation of Temperature Effects on Double-Layer and Charge-Transfer Processes for Platinum|Solution Interphases. Entropy of Formation of the Double Layer and Absolute Molar Entropy of Adsorbed Hydrogen and OH on Pt(111)

The present work combines results from pulsed-laser experiments and a thermodynamic analysis, to separate temperature effects on double-layer and charge-transfer processes. The entropy of formation of the double layer for Pt(111) in 0.1 M HClO4 is evaluated and compared with the corresponding entropy of formation of the overall interphase. The difference between the two quantities is due to the entropy change associated with charge-transfer processes, thus allowing for the evaluation of the absolute molar entropy of adsorbed hydrogen and OH on Pt(111). The results indicate that adsorbed hydrogen on Pt(111) is very mobile, whereas adsorbed OH species are rather immobile. Finally, values of the entropy of formation of the double layer are compared to data for mercury electrodes. It is concluded that the entropy of the interfacial water network on Pt(111) exhibits a nearly symmetrical dependence on charge density. This result contrasts with the behavior observed for mercury electrodes, for which the state of maximum entropy is achieved at clearly negative charge densities.

Toward a thermodynamic description of supercontinuum generation

We consider the incoherent nonlinear regime of the supercontinuum generation process in optical fibers. We show that, under certain conditions, the phenomenon of spectral broadening inherent to the supercontinuum generation may be described by simple thermodynamic arguments based on the kinetic wave theory. Accordingly, the supercontinuum generation process may be regarded as a thermalization process, which is characterized by an irreversible evolution of the optical field toward a thermodynamic equilibrium state, i.e., the state of maximum nonequilibrium entropy.

Entropy, Closures and Subgrid Modeling

Maximum entropy states or statistical mechanical equilibrium solutions have played an important role in the development of a fundamental understanding of turbulence and its role in geophysical flows. In modern general circulation models of the earth’s atmosphere and oceans most parameterizations of the subgrid-scale energy and enstrophy transfers are based on ad hoc methods or ideas developed from equilibrium statistical mechanics or entropy production hypotheses. In this paper we review recent developments in nonequilibrium statistical dynamical closure theory, its application to subgrid-scale modeling of eddy-eddy, eddy-mean field and eddy-topographic interactions and the relationship to minimum enstrophy, maximum entropy and entropy production arguments.

Spontaneous polarization induced by natural thermalization of incoherent light

We analyze theoretically the polarization properties of a partially coherent optical field that propagates in a nonlinear Kerr medium. We consider the standard model of two resonantly coupled nonlinear Schrödinger equations, which account for a wave-vector mismatch between the orthogonal polarization components. We show that such a phase-mismatch is responsible for the existence of a spontaneous repolarization process of the partially incoherent optical field during its nonlinear propagation. The repolarization process is characterized by an irreversible evolution of the unpolarized beam towards a highly polarized state, without any loss of energy. This unexpected result contrasts with the commonly accepted idea that an optical field undergoes a depolarization process under nonlinear evolution. The repolarization effect can be described in details by simple thermodynamic arguments based on the kinetic wave theory: It is shown to result from the natural tendency of the optical field to approach its thermal equilibrium state. The theory then reveals that it is thermodynamically advantageous for the optical field to evolve towards a highly polarized state, because this permits the optical field to reach the ???most disordered state???, i.e., the state of maximum (nonequilibrium) entropy. The theory is in quantitative agreement with the numerical simulations, without adjustable parameters. The physics underlying the reversible property of the repolarization process is briefly discussed in analogy with the celebrated Joule???s experiment of free expansion of a gas. Besides its fundamental interest, the repolarization effect may be exploited to achieve complete polarization of unpolarized incoherent light without loss of energy.