Entropy

Editorial

Jump to: Research, Review

4 pages, 173 KiB

Open AccessEditorial

Thermodynamics and Statistical Mechanics of Small Systems

by Andrea Puglisi, Alessandro Sarracino and Angelo Vulpiani

Entropy 2018, 20(6), 392; https://doi.org/10.3390/e20060392 - 23 May 2018

Cited by 8 | Viewed by 3406

Abstract

A challenging frontier in modern statistical physics is concerned with systems with a small number of degrees of freedom, far from the thermodynamic limit.[...] Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

Research

Jump to: Editorial, Review

21 pages, 403 KiB

Open AccessArticle

Exact Expressions of Spin-Spin Correlation Functions of the Two-Dimensional Rectangular Ising Model on a Finite Lattice

by Tao Mei

Entropy 2018, 20(4), 277; https://doi.org/10.3390/e20040277 - 12 Apr 2018

Cited by 1 | Viewed by 3860

Abstract

We employ the spinor analysis method to evaluate exact expressions of spin-spin correlation functions of the two-dimensional rectangular Ising model on a finite lattice, special process enables us to actually carry out the calculation process. We first present some exact expressions of correlation [...] Read more.

We employ the spinor analysis method to evaluate exact expressions of spin-spin correlation functions of the two-dimensional rectangular Ising model on a finite lattice, special process enables us to actually carry out the calculation process. We first present some exact expressions of correlation functions of the model with periodic-periodic boundary conditions on a finite lattice. The corresponding forms in the thermodynamic limit are presented, which show the short-range order. Then, we present the exact expression of the correlation function of the two farthest pair of spins in a column of the model with periodic-free boundary conditions on a finite lattice. Again, the corresponding form in the thermodynamic limit is discussed, from which the long-range order clearly emerges as the temperature decreases. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

10 pages, 1678 KiB

Open AccessArticle

Information Dynamics of a Nonlinear Stochastic Nanopore System

by Claire Gilpin, David Darmon, Zuzanna Siwy and Craig Martens

Entropy 2018, 20(4), 221; https://doi.org/10.3390/e20040221 - 23 Mar 2018

Cited by 4 | Viewed by 4802

Abstract

Nanopores have become a subject of interest in the scientific community due to their potential uses in nanometer-scale laboratory and research applications, including infectious disease diagnostics and DNA sequencing. Additionally, they display behavioral similarity to molecular and cellular scale physiological processes. Recent advances [...] Read more.

Nanopores have become a subject of interest in the scientific community due to their potential uses in nanometer-scale laboratory and research applications, including infectious disease diagnostics and DNA sequencing. Additionally, they display behavioral similarity to molecular and cellular scale physiological processes. Recent advances in information theory have made it possible to probe the information dynamics of nonlinear stochastic dynamical systems, such as autonomously fluctuating nanopore systems, which has enhanced our understanding of the physical systems they model. We present the results of local (LER) and specific entropy rate (SER) computations from a simulation study of an autonomously fluctuating nanopore system. We learn that both metrics show increases that correspond to fluctuations in the nanopore current, indicating fundamental changes in information generation surrounding these fluctuations. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

Figure 1
Potential at fixed positive values of y (A) and fixed negative values of y (B). The <math display="inline"> <semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> configuration is shown on both plots. As y becomes more positive, it causes the potential to skew towards a transition to negative x. As y becomes more negative, it causes the potential to skew towards a transition to positive x. Physically, positive values of x in this graph correspond to positive current values. Full article ">Figure 2
Top: the nanopore current, with each orange dot representing a measurement. Middle: the estimate of the local entropy rate (LER) of the nanopore system as a function of time. Bottom: the estimate of the specific entropy rate (SER) of the nanopore system as a function of time. This is a representative excerpt from a 40,000 point time series containing on the order of 100 transitions. Full article ">Figure 3
A projection of the reconstructed state space for the nanopore system shaded by the estimates of the LER (left) and SER (right) associated with the overall state. The plots reveal a clear trajectory in the reconstructed state space, and the arrows indicate the direction along the transitions between open and closed states. Along this trajectory, regions of relatively low surprise (LER) and low uncertainty (SER) occur when the system is in an open/closed state. Conversely, in the central regions, corresponding to transitions, we see increases in both the LER and SER. Anti-symmetry is noted in the onset of increase in SER, which shows that uncertainty is highest at the beginning of a transition and decreases as the transition proceeds to completion. (a) LER; (b) SER. Full article ">Figure 4
A 2D cross section of the reconstructed state space constructed from points within <math display="inline"> <semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mo>±</mo> <mspace width="3.33333pt"/> <mn>0.05</mn> </mrow> </semantics> </math> of the <math display="inline"> <semantics> <mrow> <msub> <mi>x</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> (left) and <math display="inline"> <semantics> <mrow> <msub> <mi>x</mi> <mrow> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> (right) planes for each plot, respectively. These plots show that information is generated most heavily around atypical transition events (the highest LER visible on the periphery of the transition tubes in the LER plot), and there is relatively uniform, high uncertainty for all transitions in the SER plot. (a) LER; (b) SER. Full article ">

768 KiB

Open AccessArticle

Information Landscape and Flux, Mutual Information Rate Decomposition and Connections to Entropy Production

by Qian Zeng and Jin Wang

Entropy 2017, 19(12), 678; https://doi.org/10.3390/e19120678 - 11 Dec 2017

Cited by 12 | Viewed by 4140

Abstract

We explored the dynamics of two interacting information systems. We show that for the Markovian marginal systems, the driving force for information dynamics is determined by both the information landscape and information flux. While the information landscape can be used to construct the [...] Read more.

We explored the dynamics of two interacting information systems. We show that for the Markovian marginal systems, the driving force for information dynamics is determined by both the information landscape and information flux. While the information landscape can be used to construct the driving force to describe the equilibrium time-reversible information system dynamics, the information flux can be used to describe the nonequilibrium time-irreversible behaviors of the information system dynamics. The information flux explicitly breaks the detailed balance and is a direct measure of the degree of the nonequilibrium or time-irreversibility. We further demonstrate that the mutual information rate between the two subsystems can be decomposed into the equilibrium time-reversible and nonequilibrium time-irreversible parts, respectively. This decomposition of the Mutual Information Rate (MIR) corresponds to the information landscape-flux decomposition explicitly when the two subsystems behave as Markov chains. Finally, we uncover the intimate relationship between the nonequilibrium thermodynamics in terms of the entropy production rates and the time-irreversible part of the mutual information rate. We found that this relationship and MIR decomposition still hold for the more general stationary and ergodic cases. We demonstrate the above features with two examples of the bivariate Markov chains. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

593 KiB

Open AccessArticle

Magnetic Engine for the Single-Particle Landau Problem

by Francisco J. Peña, Alejandro González, Alvaro S. Nunez, Pedro A. Orellana, René G. Rojas and Patricio Vargas

Entropy 2017, 19(12), 639; https://doi.org/10.3390/e19120639 - 25 Nov 2017

Cited by 13 | Viewed by 5195

Abstract

We study the effect of the degeneracy factor in the energy levels of the well-known Landau problem for a magnetic engine. The scheme of the cycle is composed of two adiabatic processes and two isomagnetic processes, driven by a quasi-static modulation of external [...] Read more.

We study the effect of the degeneracy factor in the energy levels of the well-known Landau problem for a magnetic engine. The scheme of the cycle is composed of two adiabatic processes and two isomagnetic processes, driven by a quasi-static modulation of external magnetic field intensity. We derive the analytical expression of the relation between the magnetic field and temperature along the adiabatic process and, in particular, reproduce the expression for the efficiency as a function of the compression ratio. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

275 KiB

Open AccessArticle

On Work and Heat in Time-Dependent Strong Coupling

by Erik Aurell

Entropy 2017, 19(11), 595; https://doi.org/10.3390/e19110595 - 7 Nov 2017

Cited by 24 | Viewed by 3932

Abstract

This paper revisits the classical problem of representing a thermal bath interacting with a system as a large collection of harmonic oscillators initially in thermal equilibrium. As is well known, the system then obeys an equation, which in the bulk and in the [...] Read more.

This paper revisits the classical problem of representing a thermal bath interacting with a system as a large collection of harmonic oscillators initially in thermal equilibrium. As is well known, the system then obeys an equation, which in the bulk and in the suitable limit tends to the Kramers–Langevin equation of physical kinetics. I consider time-dependent system-bath coupling and show that this leads to an additional harmonic force acting on the system. When the coupling is switched on and switched off rapidly, the force has delta-function support at the initial and final time. I further show that the work and heat functionals as recently defined in stochastic thermodynamics at strong coupling contain additional terms depending on the time derivative of the system-bath coupling. I discuss these terms and show that while they can be very large if the system-bath coupling changes quickly, they only give a finite contribution to the work that enters in Jarzynski’s equality. I also discuss that these corrections to standard work and heat functionals provide an explanation for non-standard terms in the change of the von Neumann entropy of a quantum bath interacting with a quantum system found in an earlier contribution (Aurell and Eichhorn, 2015). Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

468 KiB

Open AccessArticle

Equilibration in the Nosé–Hoover Isokinetic Ensemble: Effect of Inter-Particle Interactions

by Shamik Gupta and Stefano Ruffo

Entropy 2017, 19(10), 544; https://doi.org/10.3390/e19100544 - 14 Oct 2017

Cited by 1 | Viewed by 4336

Abstract

We investigate the stationary and dynamic properties of the celebrated Nosé–Hoover dynamics of many-body interacting Hamiltonian systems, with an emphasis on the effect of inter-particle interactions. To this end, we consider a model system with both short- and long-range interactions. The Nosé–Hoover dynamics [...] Read more.

We investigate the stationary and dynamic properties of the celebrated Nosé–Hoover dynamics of many-body interacting Hamiltonian systems, with an emphasis on the effect of inter-particle interactions. To this end, we consider a model system with both short- and long-range interactions. The Nosé–Hoover dynamics aim to generate the canonical equilibrium distribution of a system at a desired temperature by employing a set of time-reversible, deterministic equations of motion. A signature of canonical equilibrium is a single-particle momentum distribution that is Gaussian. We find that the equilibrium properties of the system within the Nosé–Hoover dynamics coincides with that within the canonical ensemble. Moreover, starting from out-of-equilibrium initial conditions, the average kinetic energy of the system relaxes to its target value over a size-independent timescale. However, quite surprisingly, our results indicate that under the same conditions and with only long-range interactions present in the system, the momentum distribution relaxes to its Gaussian form in equilibrium over a scale that diverges with the system size. On adding short-range interactions, the relaxation is found to occur over a timescale that has a much weaker dependence on system size. This system-size dependence of the timescale vanishes when only short-range interactions are present in the system. An implication of such an ultra-slow relaxation when only long-range interactions are present in the system is that macroscopic observables other than the average kinetic energy when estimated in the Nosé–Hoover dynamics may take an unusually long time to relax to its canonical equilibrium value. Our work underlines the crucial role that interactions play in deciding the equivalence between Nosé–Hoover and canonical equilibrium. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

4976 KiB

Open AccessFeature PaperArticle

Kovacs-Like Memory Effect in Athermal Systems: Linear Response Analysis

by Carlos A. Plata and Antonio Prados

Entropy 2017, 19(10), 539; https://doi.org/10.3390/e19100539 - 13 Oct 2017

Cited by 12 | Viewed by 4926

Abstract

We analyze the emergence of Kovacs-like memory effects in athermal systems within the linear response regime. This is done by starting from both the master equation for the probability distribution and the equations for the physically-relevant moments. The general results are applied to [...] Read more.

We analyze the emergence of Kovacs-like memory effects in athermal systems within the linear response regime. This is done by starting from both the master equation for the probability distribution and the equations for the physically-relevant moments. The general results are applied to a general class of models with conserved momentum and non-conserved energy. Our theoretical predictions, obtained within the first Sonine approximation, show an excellent agreement with the numerical results. Furthermore, we prove that the observed non-monotonic relaxation is consistent with the monotonic decay of the non-equilibrium entropy. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

773 KiB

Open AccessArticle

Hydrodynamics of a Granular Gas in a Heterogeneous Environment

by Francisco Vega Reyes and Antonio Lasanta

Entropy 2017, 19(10), 536; https://doi.org/10.3390/e19100536 - 11 Oct 2017

Cited by 5 | Viewed by 3746

Abstract

We analyze the transport properties of a low density ensemble of identical macroscopic particles immersed in an active fluid. The particles are modeled as inelastic hard spheres (granular gas). The non-homogeneous active fluid is modeled by means of a non-uniform stochastic thermostat. The [...] Read more.

We analyze the transport properties of a low density ensemble of identical macroscopic particles immersed in an active fluid. The particles are modeled as inelastic hard spheres (granular gas). The non-homogeneous active fluid is modeled by means of a non-uniform stochastic thermostat. The theoretical results are validated with a numerical solution of the corresponding the kinetic equation (direct simulation Monte Carlo method). We show a steady flow in the system that is accurately described by Navier-Stokes (NS) hydrodynamics, even for high inelasticity. Surprisingly, we find that the deviations from NS hydrodynamics for this flow are stronger as the inelasticity decreases. The active fluid action is modeled here with a non-uniform fluctuating volume force. This is a relevant result given that hydrodynamics of particles in complex environments, such as biological crowded environments, is still a question under intense debate. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

339 KiB

Open AccessArticle

Participation Ratio for Constraint-Driven Condensation with Superextensive Mass

by Giacomo Gradenigo and Eric Bertin

Entropy 2017, 19(10), 517; https://doi.org/10.3390/e19100517 - 26 Sep 2017

Cited by 14 | Viewed by 3532

Abstract

Broadly distributed random variables with a power-law distribution

f (m) \sim m^{- (1 + α)}

are known to generate condensation effects. This means that, when the exponent

α

lies in a certain interval, the largest variable in a [...] Read more.

Broadly distributed random variables with a power-law distribution

f (m) \sim m^{- (1 + α)}

are known to generate condensation effects. This means that, when the exponent

α

lies in a certain interval, the largest variable in a sum of N (independent and identically distributed) terms is for large N of the same order as the sum itself. In particular, when the distribution has infinite mean (

0 < α < 1

) one finds unconstrained condensation, whereas for

α > 1

constrained condensation takes places fixing the total mass to a large enough value

M = \sum_{i = 1}^{N} m_{i} > M_{c}

. In both cases, a standard indicator of the condensation phenomenon is the participation ratio

Y_{k} = 〈 \sum_{i} m_{i}^{k} / {(\sum_{i} m_{i})}^{k} 〉

(

k > 1

), which takes a finite value for

N \to \infty

when condensation occurs. To better understand the connection between constrained and unconstrained condensation, we study here the situation when the total mass is fixed to a superextensive value

M \sim N^{1 + δ}

(

δ > 0

), hence interpolating between the unconstrained condensation case (where the typical value of the total mass scales as

M \sim N^{1 / α}

for

α < 1

) and the extensive constrained mass. In particular we show that for exponents

α < 1

a condensate phase for values

δ > δ_{c} = 1 / α - 1

is separated from a homogeneous phase at

δ < δ_{c}

from a transition line,

δ = δ_{c}

, where a weak condensation phenomenon takes place. We focus on the evaluation of the participation ratio as a generic indicator of condensation, also recalling or presenting results in the standard cases of unconstrained mass and of fixed extensive mass. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

794 KiB

Open AccessArticle

Far-From-Equilibrium Time Evolution between Two Gamma Distributions

by Eun-jin Kim, Lucille-Marie Tenkès, Rainer Hollerbach and Ovidiu Radulescu

Entropy 2017, 19(10), 511; https://doi.org/10.3390/e19100511 - 22 Sep 2017

Cited by 14 | Viewed by 5162

Abstract

Many systems in nature and laboratories are far from equilibrium and exhibit significant fluctuations, invalidating the key assumptions of small fluctuations and short memory time in or near equilibrium. A full knowledge of Probability Distribution Functions (PDFs), especially time-dependent PDFs, becomes essential in [...] Read more.

Many systems in nature and laboratories are far from equilibrium and exhibit significant fluctuations, invalidating the key assumptions of small fluctuations and short memory time in or near equilibrium. A full knowledge of Probability Distribution Functions (PDFs), especially time-dependent PDFs, becomes essential in understanding far-from-equilibrium processes. We consider a stochastic logistic model with multiplicative noise, which has gamma distributions as stationary PDFs. We numerically solve the transient relaxation problem and show that as the strength of the stochastic noise increases, the time-dependent PDFs increasingly deviate from gamma distributions. For sufficiently strong noise, a transition occurs whereby the PDF never reaches a stationary state, but instead, forms a peak that becomes ever more narrowly concentrated at the origin. The addition of an arbitrarily small amount of additive noise regularizes these solutions and re-establishes the existence of stationary solutions. In addition to diagnostic quantities such as mean value, standard deviation, skewness and kurtosis, the transitions between different solutions are analysed in terms of entropy and information length, the total number of statistically-distinguishable states that a system passes through in time. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

568 KiB

Open AccessArticle

Thermodynamics of Small Magnetic Particles

by Eugenio E. Vogel, Patricio Vargas, Gonzalo Saravia, Julio Valdes, Antonio Jose Ramirez-Pastor and Paulo M. Centres

Entropy 2017, 19(9), 499; https://doi.org/10.3390/e19090499 - 15 Sep 2017

Cited by 3 | Viewed by 4862

Abstract

In the present paper, we discuss the interpretation of some of the results of the thermodynamics in the case of very small systems. Most of the usual statistical physics is done for systems with a huge number of elements in what is called [...] Read more.

In the present paper, we discuss the interpretation of some of the results of the thermodynamics in the case of very small systems. Most of the usual statistical physics is done for systems with a huge number of elements in what is called the thermodynamic limit, but not all of the approximations done for those conditions can be extended to all properties in the case of objects with less than a thousand elements. The starting point is the Ising model in two dimensions (2D) where an analytic solution exits, which allows validating the numerical techniques used in the present article. From there on, we introduce several variations bearing in mind the small systems such as the nanoscopic or even subnanoscopic particles, which are nowadays produced for several applications. Magnetization is the main property investigated aimed for two singular possible devices. The size of the systems (number of magnetic sites) is decreased so as to appreciate the departure from the results valid in the thermodynamic limit; periodic boundary conditions are eliminated to approach the reality of small particles; 1D, 2D and 3D systems are examined to appreciate the differences established by dimensionality is this small world; upon diluting the lattices, the effect of coordination number (bonding) is also explored; since the 2D Ising model is equivalent to the clock model with

q = 2

degrees of freedom, we combine previous results with the supplementary degrees of freedom coming from the variation of q up to

q = 20

. Most of the previous results are numeric; however, for the case of a very small system, we obtain the exact partition function to compare with the conclusions coming from our numerical results. Conclusions can be summarized in the following way: the laws of thermodynamics remain the same, but the interpretation of the results, averages and numerical treatments need special care for systems with less than about a thousand constituents, and this might need to be adapted for different properties or devices. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

4159 KiB

Open AccessArticle

A Chain, a Bath, a Sink, and a Wall

by Stefano Iubini, Stefano Lepri, Roberto Livi, Gian-Luca Oppo and Antonio Politi

Entropy 2017, 19(9), 445; https://doi.org/10.3390/e19090445 - 25 Aug 2017

Cited by 22 | Viewed by 4429

Abstract

We numerically investigate out-of-equilibrium stationary processes emerging in a Discrete Nonlinear Schrödinger chain in contact with a heat reservoir (a bath) at temperature

T_{L}

and a pure dissipator (a sink) acting on opposite edges. Long-time molecular-dynamics simulations are performed by evolving the [...] Read more.

We numerically investigate out-of-equilibrium stationary processes emerging in a Discrete Nonlinear Schrödinger chain in contact with a heat reservoir (a bath) at temperature

T_{L}

and a pure dissipator (a sink) acting on opposite edges. Long-time molecular-dynamics simulations are performed by evolving the equations of motion within a symplectic integration scheme. Mass and energy are steadily transported through the chain from the heat bath to the sink. We observe two different regimes. For small heat-bath temperatures

T_{L}

and chemical-potentials, temperature profiles across the chain display a non-monotonous shape, remain remarkably smooth and even enter the region of negative absolute temperatures. For larger temperatures

T_{L}

, the transport of energy is strongly inhibited by the spontaneous emergence of discrete breathers, which act as a thermal wall. A strongly intermittent energy flux is also observed, due to the irregular birth and death of breathers. The corresponding statistics exhibit the typical signature of rare events of processes with large deviations. In particular, the breather lifetime is found to be ruled by a stretched-exponential law. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

777 KiB

Open AccessArticle

Parameterization of Coarse-Grained Molecular Interactions through Potential of Mean Force Calculations and Cluster Expansion Techniques

by Anastasios Tsourtis, Vagelis Harmandaris and Dimitrios Tsagkarogiannis

Entropy 2017, 19(8), 395; https://doi.org/10.3390/e19080395 - 1 Aug 2017

Cited by 17 | Viewed by 5999

Abstract

We present a systematic coarse-graining (CG) strategy for many particle molecular systems based on cluster expansion techniques. We construct a hierarchy of coarse-grained Hamiltonians with interaction potentials consisting of two, three and higher body interactions. In this way, the suggested model becomes computationally [...] Read more.

We present a systematic coarse-graining (CG) strategy for many particle molecular systems based on cluster expansion techniques. We construct a hierarchy of coarse-grained Hamiltonians with interaction potentials consisting of two, three and higher body interactions. In this way, the suggested model becomes computationally tractable, since no information from long n-body (bulk) simulations is required in order to develop it, while retaining the fluctuations at the coarse-grained level. The accuracy of the derived cluster expansion based on interatomic potentials is examined over a range of various temperatures and densities and compared to direct computation of the pair potential of mean force. The comparison of the coarse-grained simulations is done on the basis of the structural properties, against detailed all-atom data. On the other hand, by construction, the approximate coarse-grained models retain, in principle, the thermodynamic properties of the atomistic model without the need for any further parameter fitting. We give specific examples for methane and ethane molecules in which the coarse-grained variable is the centre of mass of the molecule. We investigate different temperature (T) and density (

ρ

) regimes, and we examine differences between the methane and ethane systems. Results show that the cluster expansion formalism can be used in order to provide accurate effective pair and three-body CG potentials at high T and low

ρ

regimes. In the liquid regime, the three-body effective CG potentials give a small improvement over the typical pair CG ones; however, in order to get significantly better results, one needs to consider even higher order terms. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

550 KiB

Open AccessArticle

An Application of Pontryagin’s Principle to Brownian Particle Engineered Equilibration

by Paolo Muratore-Ginanneschi and Kay Schwieger

Entropy 2017, 19(7), 379; https://doi.org/10.3390/e19070379 - 24 Jul 2017

Cited by 10 | Viewed by 4334

Abstract

We present a stylized model of controlled equilibration of a small system in a fluctuating environment. We derive the optimal control equations steering in finite-time the system between two equilibrium states. The corresponding thermodynamic transition is optimal in the sense that it occurs [...] Read more.

We present a stylized model of controlled equilibration of a small system in a fluctuating environment. We derive the optimal control equations steering in finite-time the system between two equilibrium states. The corresponding thermodynamic transition is optimal in the sense that it occurs at minimum entropy if the set of admissible controls is restricted by certain bounds on the time derivatives of the protocols. We apply our equations to the engineered equilibration of an optical trap considered in a recent proof of principle experiment. We also analyze an elementary model of nucleation previously considered by Landauer to discuss the thermodynamic cost of one bit of information erasure. We expect our model to be a useful benchmark for experiment design as it exhibits the same integrability properties of well-known models of optimal mass transport by a compressible velocity field. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

269 KiB

Open AccessFeature PaperArticle

Clausius Relation for Active Particles: What Can We Learn from Fluctuations

by Andrea Puglisi and Umberto Marini Bettolo Marconi

Entropy 2017, 19(7), 356; https://doi.org/10.3390/e19070356 - 13 Jul 2017

Cited by 44 | Viewed by 5086

Abstract

Many kinds of active particles, such as bacteria or active colloids, move in a thermostatted fluid by means of self-propulsion. Energy injected by such a non-equilibrium force is eventually dissipated as heat in the thermostat. Since thermal fluctuations are much faster and weaker [...] Read more.

Many kinds of active particles, such as bacteria or active colloids, move in a thermostatted fluid by means of self-propulsion. Energy injected by such a non-equilibrium force is eventually dissipated as heat in the thermostat. Since thermal fluctuations are much faster and weaker than self-propulsion forces, they are often neglected, blurring the identification of dissipated heat in theoretical models. For the same reason, some freedom—or arbitrariness—appears when defining entropy production. Recently three different recipes to define heat and entropy production have been proposed for the same model where the role of self-propulsion is played by a Gaussian coloured noise. Here we compare and discuss the relation between such proposals and their physical meaning. One of these proposals takes into account the heat exchanged with a non-equilibrium active bath: such an “active heat” satisfies the original Clausius relation and can be experimentally verified. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

1313 KiB

Open AccessFeature PaperArticle

Fourier’s Law in a Generalized Piston Model

by Lorenzo Caprini, Luca Cerino, Alessandro Sarracino and Angelo Vulpiani

Entropy 2017, 19(7), 350; https://doi.org/10.3390/e19070350 - 11 Jul 2017

Cited by 6 | Viewed by 3856

Abstract

A simplified, but non trivial, mechanical model—gas of N particles of mass m in a box partitioned by n mobile adiabatic walls of mass M—interacting with two thermal baths at different temperatures, is discussed in the framework of kinetic theory. Following an [...] Read more.

A simplified, but non trivial, mechanical model—gas of N particles of mass m in a box partitioned by n mobile adiabatic walls of mass M—interacting with two thermal baths at different temperatures, is discussed in the framework of kinetic theory. Following an approach due to Smoluchowski, from an analysis of the collisions particles/walls, we derive the values of the main thermodynamic quantities for the stationary non-equilibrium states. The results are compared with extensive numerical simulations; in the limit of large n,

m N / M ≫ 1

and

m / M ≪ 1

, we find a good approximation of Fourier’s law. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

1575 KiB

Open AccessArticle

Information-Theoretic Bound on the Entropy Production to Maintain a Classical Nonequilibrium Distribution Using Ancillary Control

by Jordan M. Horowitz and Jeremey L. England

Entropy 2017, 19(7), 333; https://doi.org/10.3390/e19070333 - 4 Jul 2017

Cited by 8 | Viewed by 4224

Abstract

There are many functional contexts where it is desirable to maintain a mesoscopic system in a nonequilibrium state. However, such control requires an inherent energy dissipation. In this article, we unify and extend a number of works on the minimum energetic cost to [...] Read more.

There are many functional contexts where it is desirable to maintain a mesoscopic system in a nonequilibrium state. However, such control requires an inherent energy dissipation. In this article, we unify and extend a number of works on the minimum energetic cost to maintain a mesoscopic system in a prescribed nonequilibrium distribution using ancillary control. For a variety of control mechanisms, we find that the minimum amount of energy dissipation necessary can be cast as an information-theoretic measure of distinguishability between the target nonequilibrium state and the underlying equilibrium distribution. This work offers quantitative insight into the intuitive idea that more energy is needed to maintain a system farther from equilibrium. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

Figure 1
Illustration of three types of control: (Top) Graph representation of the system’s configuration space without control. Mesoscopic configurations are represented as vertices (or nodes) with edges signifying allowed transitions; (Bottom) Control is implemented by adding additional edges (red dashed) or nodes (red dots) in order to drive the system into a nonequilibrium distribution. From Left to Right: Edge control, Node control, and Auxiliary control. Full article ">Figure 2
Illustration of chemical control: Two species <math display="inline"> <semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics> </math> and <math display="inline"> <semantics> <msub> <mi>X</mi> <mn>2</mn> </msub> </semantics> </math> interconvert through a single chemical reaction <math display="inline"> <semantics> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>⇌</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> </mrow> </semantics> </math> that conserves particle number, depicted as solid black lines. As a result, the dynamical evolution of this chemical reaction network is restricted to a single diagonal subspace of the network of states. Control can be implemented by chemostating one of the species, say <math display="inline"> <semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics> </math>, by allowing <math display="inline"> <semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics> </math> molecules to be added and subtracted from the reaction volume through the reaction <math display="inline"> <semantics> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>⇌</mo> <mi>ϕ</mi> </mrow> </semantics> </math>, depicted as red dashed lines. As this reaction breaks the particle number conservation law, it extends the possible configurations the system can dynamically explore. Full article ">

545 KiB

Open AccessArticle

Lyapunov Spectra of Coulombic and Gravitational Periodic Systems

by Pankaj Kumar and Bruce N. Miller

Entropy 2017, 19(5), 238; https://doi.org/10.3390/e19050238 - 20 May 2017

Cited by 3 | Viewed by 5167

Abstract

An open question in nonlinear dynamics is the relation between the Kolmogorov entropy and the largest Lyapunov exponent of a given orbit. Both have been shown to have diagnostic capability for phase transitions in thermodynamic systems. For systems with long-range interactions, the choice [...] Read more.

An open question in nonlinear dynamics is the relation between the Kolmogorov entropy and the largest Lyapunov exponent of a given orbit. Both have been shown to have diagnostic capability for phase transitions in thermodynamic systems. For systems with long-range interactions, the choice of boundary plays a critical role and appropriate boundary conditions must be invoked. In this work, we compute Lyapunov spectra for Coulombic and gravitational versions of the one-dimensional systems of parallel sheets with periodic boundary conditions. Exact expressions for time evolution of the tangent-space vectors are derived and are utilized toward computing Lypaunov characteristic exponents using an event-driven algorithm. The results indicate that the energy dependence of the largest Lyapunov exponent emulates that of Kolmogorov entropy for each system for a given system size. Our approach forms an effective and approximation-free instrument for studying the dynamical properties exhibited by the Coulombic and gravitational systems and finds applications in investigating indications of thermodynamic transitions in small as well as large versions of the spatially periodic systems. When a phase transition exists, we find that the largest Lyapunov exponent serves as a precursor of the transition that becomes more pronounced as the system size increases. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

301 KiB

Open AccessArticle

Stochastic Stirling Engine Operating in Contact with Active Baths

by Ruben Zakine, Alexandre Solon, Todd Gingrich and Frédéric Van Wijland

Entropy 2017, 19(5), 193; https://doi.org/10.3390/e19050193 - 27 Apr 2017

Cited by 61 | Viewed by 10158

Abstract

A Stirling engine made of a colloidal particle in contact with a nonequilibrium bath is considered and analyzed with the tools of stochastic energetics. We model the bath by non Gaussian persistent noise acting on the colloidal particle. Depending on the chosen definition [...] Read more.

A Stirling engine made of a colloidal particle in contact with a nonequilibrium bath is considered and analyzed with the tools of stochastic energetics. We model the bath by non Gaussian persistent noise acting on the colloidal particle. Depending on the chosen definition of an isothermal transformation in this nonequilibrium setting, we find that either the energetics of the engine parallels that of its equilibrium counterpart or, in the simplest case, that it ends up being less efficient. Persistence, more than non-Gaussian effects, are responsible for this result. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

Figure 1
Schematic diagram of the Stirling cycle in stiffness-position space. Unlike its thermodynamic counterpart, the cycle is run counter-clockwise but is nevertheless an engine cycle. Full article ">Figure 2
Log of the probability of the colloid’s position as a function of position (for a unit a), in equilibrium with Gaussian statistics (red at <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>/</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math>, green at <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>/</mo> <mi>k</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math>) or out of equilibrium as given by <math display="inline"> <semantics> <msub> <mi>P</mi> <mi>ss</mi> </msub> </semantics> </math> (blue at <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>/</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math>, orange at <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>/</mo> <mi>k</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math>). Full article ">

Review

Jump to: Editorial, Research

16 pages, 899 KiB

Open AccessReview

Fluctuations, Finite-Size Effects and the Thermodynamic Limit in Computer Simulations: Revisiting the Spatial Block Analysis Method

by Maziar Heidari, Kurt Kremer, Raffaello Potestio and Robinson Cortes-Huerto

Entropy 2018, 20(4), 222; https://doi.org/10.3390/e20040222 - 24 Mar 2018

Cited by 27 | Viewed by 6155

Abstract

The spatial block analysis (SBA) method has been introduced to efficiently extrapolate thermodynamic quantities from finite-size computer simulations of a large variety of physical systems. In the particular case of simple liquids and liquid mixtures, by subdividing the simulation box into blocks of [...] Read more.

The spatial block analysis (SBA) method has been introduced to efficiently extrapolate thermodynamic quantities from finite-size computer simulations of a large variety of physical systems. In the particular case of simple liquids and liquid mixtures, by subdividing the simulation box into blocks of increasing size and calculating volume-dependent fluctuations of the number of particles, it is possible to extrapolate the bulk isothermal compressibility and Kirkwood–Buff integrals in the thermodynamic limit. Only by explicitly including finite-size effects, ubiquitous in computer simulations, into the SBA method, the extrapolation to the thermodynamic limit can be achieved. In this review, we discuss two of these finite-size effects in the context of the SBA method due to (i) the statistical ensemble and (ii) the finite integration domains used in computer simulations. To illustrate the method, we consider prototypical liquids and liquid mixtures described by truncated and shifted Lennard–Jones (TSLJ) potentials. Furthermore, we show some of the most recent developments of the SBA method, in particular its use to calculate chemical potentials of liquids in a wide range of density/concentration conditions. Full article

(This article belongs to the Special Issue Thermodynamics and Statistical Mechanics of Small Systems)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Thermodynamics and Statistical Mechanics of Small Systems

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Published Papers (21 papers)

Editorial

Research

Review

Further Information

Guidelines

MDPI Initiatives

Follow MDPI