On the problem of the metastable region at morphological instability

The study deals with numerical analysis of the morphological stability of a growing round particle with respect to harmonic perturbations of an arbitrary amplitude. Various growth regimes (from diffusion to kinetic-limited) are considered. It is found that the critical size of the particle stability decreases as the perturbation amplitude increases and tends to the value, which was determined analytically elsewhere using the maximum entropy production principle. This result is a crucial argument in support of the hypothesis that the entropy production can be used for analysis of a nonequilibrium phase transitions similarly to thermodynamic potentials in the case of equilibrium phase transitions.     

Time Dependence of Joint Entropy of Oscillating Quantum Systems

The time dependent entropy (or Leipnik’s entropy) of harmonic and damped harmonic oscillator systems is studied by using time dependent wave function obtained by the Feynman path integral method. The Leipnik entropy and its envelope change as a function of time, angular frequency and damping factor. Our results for simple harmonic oscillator are in agreement with the literature. However, the joint entropy of damped harmonic oscillator shows remarkable discontinuity with time for certain values of damping factor. The envelope of the joint entropy curve increases with time monotonically. These results show the general properties of the envelope of the joint entropy curve for quantum systems.     

Onsager-Machlup Theory and Work Fluctuation Theorem for a Harmonically Driven Brownian Particle

We extend Tooru-Cohen analysis for nonequilibrium steady state (NSS) of a Brownian particle to nonequilibrium oscillatory state (NOS) of Brownian particle by considering time dependent external drive protocol. We consider an unbounded charged Brownian particle in the presence of oscillating electric field and prove work fluctuation theorem, which is valid for any initial distribution and at all times. For harmonically bounded and constantly dragged Brownian particle considered by Tooru and Cohen, work fluctuation theorem is valid for any initial condition (also NSS), but only in large time limit. We use Onsager-Machlup Lagrangian with a constraint to obtain frequency dependent work distribution function, and describe entropy production rate and properties of dissipation functions for the present system using Onsager-Machlup functional.     

Entropy identity and equilibrium conditions in relativistic thermodynamics

Starting out with the balance equations for energy-momentum, spin, particle and entropy density, an approach is considered which represents a framework for special- and general-relativistic continuum thermodynamics. A general entropy density 4-vector, containing particle, energy-momentum, and spin density contributions, is introduced. This makes possible, firstly, to test special entropy density 4-vectors used by other authors with respect to their generality and validity and, secondly, to determine entropy supply and entropy production. Using this entropy density 4-vector, material-independent equilibrium conditions are discussed. While in literature, generally thermodynamic equilibrium is determined by introducing a variety of conditions by hand, the present approach proceeds as follows: For a comparatively wide class of space–time geometries, the necessary equilibrium conditions of vanishing entropy supply and vanishing entropy production are exploited. Because these necessary equilibrium conditions do not determine the equilibrium, supplementary conditions are added systematically motivated by the requirement that also all parts of the necessary conditions have to be fixed in equilibrium.     

Synthesis of Two-Color Laser Pulses for the Harmonic Cutoff Extension

Increasing simultaneously both the cutoff energy and efficiency is a big challenge to all applications of high-order harmonic generation(HHG).For this purpose,the shaping of the waveform of driving pulse is an alternative approach.Here,we show that the harmonic cutoff can be extended by about two times without reducing harmonic yield after considering macroscopic propagation effects,by adopting a practical way to synthesize two-color fields with fixed energy.Our results,combined with the experimental techniques,show the great potential of HHG as a tabletop light source.     

Entropy as a measure of diffusion

The time variation of entropy, as an alternative to the variance, is proposed as a measure of the diffusion rate. It is shown that for linear and time-translationally invariant systems having a large-time limit for the density, at large times the entropy tends exponentially to a constant. For systems with no stationary density, at large times the entropy is logarithmic with a coefficient specifying the speed of the diffusion. As an example, the large-time behaviors of the entropy and the variance are compared for various types of fractional-derivative diffusions.     

Quantum HeisenbergS=1 spin glass: Effect of anisotropy and ferromagnetic interaction

A complete analysis of the transfer dynamics in an asymmetric nonlinear dimer model with different cubic site polarizations is given. The analysis is performed for both the dynamics of the full density matrix on the Bloch sphere (location of fixed points, bifurcation in dependence on the polarization strength) and of the reduced space of the occupation difference using a potential function. For a time dependent harmonic perturbation the appearance of chaotic transfer regimes near a homoclinic structure on the Bloch sphere is demonstrated. A comparison with spin models is performed. It is shown that the chaotic regime corresponds to chaotic motion in a classical spin model witha–S _z ² nonlinearity and an external magnetic field having its constant and time dependent parts in the same direction.     

Work distribution function for a Brownian particle driven by a nonconservative force

We derive the distribution function of work performed by a harmonic force acting on a uniformly dragged Brownian particle subjected to a rotational torque. Following the Onsager and Machlup’s functional integral approach, we obtain the transition probability of finding the Brownian particle at a particular position at time t given that it started the journey from a specific location at an earlier time. The difference between the forward and the time-reversed form of the generalized Onsager-Machlup’s Lagrangian is identified as the rate of medium entropy production which further helps us develop the stochastic thermodynamics formalism for our model. The probability distribution for the work done by the harmonic trap is evaluated for an equilibrium initial condition. Although this distribution has a Gaussian form, it is found that the distribution does not satisfy the conventional work fluctuation theorem.     

Entropy production in open volume-preserving systems

We describe a mechanism leading to positive entropy production in volume-preserving systems under nonequilibrium conditions. We consider volume-preserving systems sustaining a diffusion process like the multibaker map or the Lorentz gas. A continuous flux of particles is imposed across the system resulting in a steady gradient of concentration. In the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient, the invariant measure becomes singular. For instance, in the multibaker map, the limit invariant measure has a cumulative function given in terms of the nondifferentiable Takagi function. Because of this singularity of the invariant measure, the entropy must be defined as a coarse-grained entropy instead of the fined-grained Gibbs entropy, which would require the existence of a regular measure with a density. The coarse-grained entropy production is then shown to be asymptotically positive and, moreover, given by the entropy production expected from irreversible thermodynamics.     

Energy sharing induced by the nonlinear interaction

Strong energy sharing is shown by numerically investigating coupled multi-component Bose–Einstein condensates(BECs) with a harmonic trap to simulate the Fermi–Pasta–Ulam model(FPU). For two-component BECs, the energy exchanging between each part, from regular, quantum beating to complete energy sharing, is explored by simulating their Husimi distributions, the time evolution of energies and the statistical entropy. Meanwhile, in the three-component case, a more complex energy sharing behavior is reported and a strong energy sharing is found.     

Excess Entropy Production in Quantum System: Quantum Master Equation Approach

For open systems described by the quantum master equation (QME), we investigate the excess entropy production under quasistatic operations between nonequilibrium steady states. The average entropy production is composed of the time integral of the instantaneous steady entropy production rate and the excess entropy production. We propose to define average entropy production rate using the average energy and particle currents, which are calculated by using the full counting statistics with QME. The excess entropy production is given by a line integral in the control parameter space and its integrand is called the Berry–Sinitsyn–Nemenman (BSN) vector. In the weakly nonequilibrium regime, we show that BSN vector is described by \(\ln \breve{\rho }_0\) and \(\rho _0\) where \(\rho _0\) is the instantaneous steady state of the QME and \(\breve{\rho }_0\) is that of the QME which is given by reversing the sign of the Lamb shift term. If the system Hamiltonian is non-degenerate or the Lamb shift term is negligible, the excess entropy production approximately reduces to the difference between the von Neumann entropies of the system. Additionally, we point out that the expression of the entropy production obtained in the classical Markov jump process is different from our result and show that these are approximately equivalent only in the weakly nonequilibrium regime.     

Entropy production along a stochastic trajectory and an integral fluctuation theorem

For stochastic nonequilibrium dynamics like a Langevin equation for a colloidal particle or a master equation for discrete states, entropy production along a single trajectory is studied. It involves both genuine particle entropy and entropy production in the surrounding medium. The integrated sum of both Delatas(tot) is shown to obey a fluctuation theorem (exp([-Deltas(tot) = 1 for arbitrary initial conditions and arbitrary time-dependent driving over a finite time interval.     

Entropy production, fractals, and relaxation to equilibrium

The theory of entropy production in nonequilibrium, Hamiltonian systems, previously described for steady states using partitions of phase space, is here extended to time dependent systems relaxing to equilibrium. We illustrate the main ideas by using a simple multibaker model, with some nonequilibrium initial state, and we study its progress toward equilibrium. The central results are (i) the entropy production is governed by an underlying, exponentially decaying fractal structure in phase space, (ii) the rate of entropy production is largely independent of the scale of resolution used in the partitions, and (iii) the rate of entropy production is in agreement with the predictions of nonequilibrium thermodynamics.     

Estimation of entropies and dimensions by nonlinear symbolic time series analysis

Symbolic nonlinear time series analysis methods have the potential for analyzing nonlinear data efficiently with low sensitivity to noise. In symbolic nonlinear time series analysis a time series for a fixed delay is partitioned into a small number (called the alphabet size) of cells labeled by symbols, creating a symbolic time series. Symbolic methods involve computing the statistics of words made from the symbolic time series. Specifically, the Shannon entropy of the distribution of possible words for a range of word lengths is computed. The rate of increase of the entropy with word length is the metric (Kolmogorov-Sinai) entropy. Methods of computing the metric entropy for flows as well as for maps are shown. A method of computing the information dimension appropriate to symbolic analysis is proposed. In terms of this formulation, the information dimension is determined by the scaling of entropy as alphabet size is modestly increased, using the information obtained from large word length. We discuss the role of sampling time and the issue of using these methods when there may be no generating partition.     

Hurst-Kolmogorov dynamics as a result of extremal entropy production

It is demonstrated that extremization of entropy production of stochastic representations of natural systems, performed at asymptotic times (zero or infinity) results in constant derivative of entropy in logarithmic time and, in turn, in Hurst-Kolmogorov processes. The constraints used include preservation of the mean, variance and lag-1 autocovariance at the observation time step, and an inequality relationship between conditional and unconditional entropy production, which is necessary to enable physical consistency. An example with real world data illustrates the plausibility of the findings.     

Relaxation of ideal classical particles in a one‐dimensional box

We study the deterministic dynamics of non‐interacting classical gas particles confined to a one‐dimensional box as a pedagogical toy model for the relaxation of the Boltzmann distribution towards equilibrium. Hard container walls alone induce a uniform distribution of the gas particles at large times. For the relaxation of the velocity distribution we model the dynamical walls by independent scatterers. The Markov property guarantees a stationary but not necessarily thermal velocity distribution for the gas particles at large times. We identify the conditions for physical walls where the stationary velocity distribution is the Maxwell distribution. For our numerical simulation we represent the wall particles by independent harmonic oscillators. The corresponding dynamical map for oscillators with a fixed phase (Fermi–Ulam accelerator) is chaotic for mesoscopic box dimensions.     

Performance of a quantum heat engine cycle working with harmonic oscillator systems

A cycle model of an irreversible heat engine working with harmonic systems is established in this paper. Based on the equation of motion of an operator in the Heisenberg picture and semi-group approach, the first law of thermodynamics for a harmonic system and the time evolution of the system are obtained. The general expressions for several important parameters, such as the work, efficiency, power output, and rate of entropy production are derived. By means of numerical analysis, the optimally operating regions and the optimal values of performance parameters of the cycle are determined under the condition of maximum power output. At last, some special cases, such as high temperature limit and frictionless case, are dis-cussed in brief.     

Noise and conditional entropy evolution

We study the convergence properties of the conditional (Kullback–Leibler) entropy in stochastic systems. We have proved general results showing that asymptotic stability is a necessary and sufficient condition for the monotone convergence of the conditional entropy to its maximal value of zero. Additionally we have made specific calculations of the rate of convergence of this entropy to zero in a one-dimensional situation, illustrated by Ornstein–Uhlenbeck and Rayleigh processes, higher dimensional situations, and a two-dimensional Ornstein–Uhlenbeck process with a stochastically perturbed harmonic oscillator and colored noise as examples. We also apply our general results to the problem of conditional entropy convergence in the presence of dichotomous noise. In both the one-dimensional and multidimensional cases we show that the convergence of the conditional entropy to zero is monotone and at least exponential. In the specific cases of the Ornstein–Uhlenbeck and Rayleigh processes, as well as the stochastically perturbed harmonic oscillator and colored noise examples, we obtain exact formulae for the temporal evolution of the conditional entropy starting from a concrete initial distribution. The rather surprising result in this case is that the rate of convergence of the entropy to zero is independent of the noise amplitude.     

Physical Consequences of Noncanonical Quantizations — An Example of <Emphasis Type="Italic">so</Emphasis>(2,1) Oscillator

We consider noncanonical harmonic oscillator quantized as a representation of the so(2,1) algebra and investigate its termodynamical properties. For all temperatures the entropy of such a noncanonical oscillator is greater than the entropy of the canonical one. This may serve as an explanation why in the equilibrium the oscillator is canonical.