The third law of thermodynamics and the degeneracy of the ground state for lattice systems

The third law of thermodynamics, in the sense that the entropy per unit volume goes to zero as the temperature goes to zero, is investigated within the framework of statistical mechanics for quantum and classical lattice models. We present two main results: (i) For all models the question of whether the third law is satisfied can be decided completely in terms of ground-state degeneracies alone, provided these are computed for all possible boundary conditions. In principle, there is no need to investigate possible entropy contributions from low-lying excited states, (ii) The third law is shown to hold for ferromagnetic models by an analysis of the ground states.Dedicated to Pierre Résibois. Work supported in part by NSF grant PHY-7825390 A01.     

Discovering Noncritical Organization: Statistical Mechanical,Information Theoretic,and Computational Views of Patterns in One-Dimensional Spin Systems

We compare and contrast three different, but complementary views of “structure” and “pattern” in spatial processes. For definiteness and analytical clarity, we apply all three approaches to the simplest class of spatial processes: one-dimensional Ising spin systems with finite-range interactions. These noncritical systems are well-suited for this study since the change in structure as a function of system parameters is more subtle than that found in critical systems where, at a phase transition, many observables diverge, thereby making the detection of change in structure obvious. This survey demonstrates that the measures of pattern from information theory and computational mechanics differ from known thermodynamic and statistical mechanical functions. Moreover, they capture important structural features that are otherwise missed. In particular, a type of mutual information called the excess entropy—an information theoretic measure of memory—serves to detect ordered, low entropy density patterns. It is superior in several respects to other functions used to probe structure, such as magnetization and structure factors. ϵ-Machines—the main objects of computational mechanics—are seen to be the most direct approach to revealing the (group and semigroup) symmetries possessed by the spatial patterns and to estimating the minimum amount of memory required to reproduce the configuration ensemble, a quantity known as the statistical complexity. Finally, we argue that the information theoretic and computational mechanical analyses of spatial patterns capture the intrinsic computational capabilities embedded in spin systems—how they store, transmit, and manipulate configurational information to produce spatial structure.     

Thermodynamics for Coulomb systems: A problem at vanishing particle densities

In this paper I combine techniques recently developed by Charles Fefferman with the well-known methods of Joel Lebowitz and Elliott Lieb to resolve some technical problems left unsettled by Lebowitz and Lieb's fundamental 1972 paper The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei.     

Status of the Fundamental Laws of Thermodynamics

We describe recent progress towards deriving the Fundamental Laws of thermodynamics (the 0th, 1st, and 2nd Law) from nonequilibrium quantum statistical mechanics in simple, yet physically relevant models. Along the way, we clarify some basic thermodynamic notions and discuss various reversible and irreversible thermodynamic processes from the point of view of quantum statistical mechanics.     

The quartic oscillator in an external field and the statistical physics of highly anisotropic solids

The statistical mechanics of one- and two-dimensional Ginzburg–Landau systems is evaluated analytically, via the transfer matrix method, using an expression of the ground state energy of the quartic anharmonic oscillator in an external field. In the two-dimensional case, the critical temperature of the order–disorder phase transition is expressed as a Lambert function of the inverse inter-chain coupling constant.     

关于统计力学的基本原理

文章简要讨论有关统计力学基本原理的几个问题,包括正则系综理论、系综分布函数的支集、与热力学的对应、不可逆性及分布函数的时间演化。     

Nanohybridization of Low-Dimensional Nanomaterials: Synthesis,Classification, and Application

Nanomaterials have attracted much attention from academic to industrial research. General methodologies are needed to impose architectural order in low-dimensional nanomaterials composed of nanoobjects of various shapes and sizes, such as spherical particles, rods, wires, combs, horns, and other non specified geometrical architectures. These nanomaterials are the building blocks for nanohybrid materials, whose applications have improved and will continuously enhance the quality of the daily life of mankind. In this article, we present a comprehensive review on the synthesis, dimension, properties, and present and potential future applications of nanomaterials and nanohybrids. Due to the large number of review articles on specific dimension, morphology, or application of nanomaterials, we will focus on different forms of nanomaterials, such as, linear, particulate, and miscellaneous forms. We believe that almost all the nanomaterials and nanohybrids will come under these three categories. Every form or dimension or morphology has its own significant properties and advantages. These low-dimensional nanomaterials can be integrated to create novel nano-composite material applications for next-generation devices needed to address the current energy crisis, environmental sustainability, and better performance requirements. We discuss the synthesis, properties, and morphology of different forms of nanomaterials (building blocks). Moreover, we elaborate on the synthesis, modification, and application of nanohybrids. The applications of these nanomaterials and nanohybrids in sensors, solar cells, lithium batteries, electronic, catalysis, photocatalysis, electrocatalysis, and bio-based applications will be detailed. The time is now ripe to explore new nanohybrids that use individual nanomaterial components as basic building blocks, potentially affording additionally novel behavior and leading to new, useful applications. In this regard, the combination or integration of linear nanorods/nanowires and spherical nanoparticles to produce mixed-dimensionality, higher-level nanocomposites of greater complexity is an interesting theme, which we explore in this review article.     

Thermodynamics of Self-Gravitating Systems

This work assembles some basic theoretical elements on thermal equilibrium, stability conditions, and fluctuation theory in self-gravitating systems illustrated with a few examples. Thermodynamics deals with states that have settled down after sufficient time has gone by. Time dependent phenomena are beyond the scope of this paper. While thermodynamics is firmly rooted in statistical physics, equilibrium configurations, stability criteria and the destabilizing effect of fluctuations are all expressed in terms of thermodynamic functions. The work is not a review paper but a pedagogical introduction which may interest theoreticians in astronomy and astrophysicists. It contains sufficient mathematical details for the reader to redo all calculations. References are only to seminal works or readable reviews. Delicate mathematical problems are mentioned but are not discussed in detail.     

Comment on ‘Time-dependent paths,fictive temperatures and residual entropy of glass’

In their recent paper, Johari and Aji claim that there is a positive residual entropy of a glass. Here we show that their arguments in support of this view are incorrect.     

Heat transport in low-dimensional systems

Recent results on theoretical studies of heat conduction in low-dimensional systems are presented. These studies are on simple, yet non-trivial, models. Most of these are classical systems, but some quantum-mechanical work is also reported. Much of the work has been on lattice models corresponding to phononic systems, and some on hard-particle and hard-disc systems. A recently developed approach, using generalized Langevin equations and phonon Green's functions, is explained and several applications to harmonic systems are given. For interacting systems, various analytic approaches based on the Green–Kubo formula are described, and their predictions are compared with the latest results from simulation. These results indicate that for momentum-conserving systems, transport is anomalous in one and two dimensions, and the thermal conductivity κ diverges with system size L as κ ～ L ^α. For one-dimensional interacting systems there is strong numerical evidence for a universal exponent α = 1/3, but there is no exact proof for this so far. A brief discussion of some of the experiments on heat conduction in nanowires and nanotubes is also given.     

Investigation of heat capacities of proteins by statistical mechanical methods

In this study additional heat capacity of the proteins in water dissociation have been investigated by statistical mechanical methods. For this purpose, taking electric field E$E$ and total dipole moment M$M$ as thermodynamical variables and starting with the first law of thermodynamics, an expression which reveals the thermodynamical relation between additional heat capacity in effective electric field Δ_CE$\mathrm{\Delta }{C}_E$ and additional heat capacity at the constant total dipole moment Δ_CM$\mathrm{\Delta }{C}_M$, has been obtained. It is found that, difference between the heat capacities depends linearly on temperature. To establish the hydration effect during the folding and unfolding of the proteins, physical properties of the apolar dissociation have been used [G. Oylumluoglu, et al., Physica A 361 (2006) 255–262]. In the thermodynamical investigation of the protein system, in order to introduce the chemical potential μ$\mu$ (here it takes place of pH), one has to consider the system as a macro-canonical ensemble. In this study, the macro-canonical ensemble is obtained from the canonical ensemble. In this approach the proteins are taken in a heat bath, and also it is supposed that the system is in a particle reservoir. When this reservoir reaches to an equilibrium the number of particles take an average value. In this study, with the purpose of revealing the additional effect to the heat capacity, the partition functions of the proteins obtained in single protein molecule approach are taken. In this way, additional free energy has been related to heat capacities. Calculating the heat capacity Δ_CE$\mathrm{\Delta }{C}_E$ and taking the heat capacity at constant total dipole moment Δ_CM$\mathrm{\Delta }{C}_M$ from the experimental data, the outcomes of the performed calculations have been investigated for Myoglobin and other proteins.     

Incomplete nonextensive statistics and the zeroth law of thermodynamics

On the basis of the entropy of incomplete statistics (IS) and the joint probability factorization condition, two controversial problems existing in IS are investigated: one is what expression of the internal energy is reasonable for a composite system and the other is whether the traditional zeroth law of thermodynamics is suitable for IS. Some new equivalent expressions of the internal energy of a composite system are derived through accurate mathematical calculation. Moreover, a self-consistent calculation is used to expound that the zeroth law of thermodynamics is also suitable for IS, but it cannot be proven theoretically. Finally, it is pointed out that the generalized zeroth law of thermodynamics for incomplete nonextensive statistics is unnecessary and the nonextensive assumptions for the composite internal energy will lead to mathematical contradiction.     

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.     

Thermodynamics of rotating self-gravitating systems

We investigate the statistical equilibrium properties of a system of classical particles interacting via Newtonian gravity, enclosed in a three-dimensional spherical volume. Within a mean-field approximation, we derive an equation for the density profiles maximizing the microcanonical entropy and solve it numerically. At low angular momenta, i.e. for a slowly rotating system, the well-known gravitational collapse “transition” is recovered. At higher angular momenta, instead, rotational symmetry can spontaneously break down giving rise to more complex equilibrium configurations, such as double-clusters (“double stars”). We analyze the thermodynamics of the system and the stability of the different equilibrium configurations against rotational symmetry breaking, and provide the global phase diagram. Received 8 July 2002 Published online 15 October 2002 RID="a" ID="a"e-mail: demartino@hmi.de     

简单体系温度涨落的发散问题

平衡体系热力学推导涨落的前提是涨落必须很小，如果得到一个发散的结果说明这一涨落是不可靠的.对一些体系温度涨落的热力学结果，在温度趋于绝对零度时是发散的，这时必须用统计物理来处理.对这些体系进行统计物理处理的结果表明，涨落在温度趋于绝对零度时是趋于零的. 关键词： 热力学与统计物理 涨落     

The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis

We have investigated the proof of the H theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [G. Kaniadakis, Phys. Rev. E 66, 056125 (2002); G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)]. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the Kaniadakis formalism. It is shown that the collisional equilibrium states (null entropy source term) are described by a κ power law generalization of the exponential Juttner distribution, e.g., , with θ=α(x)+β_μp^μ, where α(x) is a scalar, β_μ is a four-vector, and p^μ is the four-momentum. As a simple example, we calculate the relativistic κ power law for a dilute charged gas under the action of an electromagnetic field F^μν. All standard results are readly recovered in the particular limit κ→0.     

Quantum heat engines, the second law and Maxwell's daemon

We introduce a class of quantum heat engines which consists of two-energy-eigenstate systems, the simplest of quantum mechanical systems, undergoing quantum adiabatic processes and energy exchanges with heat baths, respectively, at different stages of a cycle. Armed with this class of heat engines and some interpretation of heat transferred and work performed at the quantum level, we are able to clarify some important aspects of the second law of thermodynamics. In particular, it is not sufficient to have the heat source hotter than the sink, but there must be a minimum temperature difference between the hotter source and the cooler sink before any work can be extracted through the engines. The size of this minimum temperature difference is dictated by that of the energy gaps of the quantum engines involved. Our new quantum heat engines also offer a practical way, as an alternative to Szilard's engine, to physically realise Maxwell's daemon. Inspired and motivated by the Rabi oscillations, we further introduce some modifications to the quantum heat engines with single-mode cavities in order to, while respecting the second law, extract more work from the heat baths than is otherwise possible in thermal equilibria. Some of the results above are also generalisable to quantum heat engines of an infinite number of energy levels including 1-D simple harmonic oscillators and 1-D infinite square wells, or even special cases of continuous spectra.     

Vasserstein distances in two-state systems

We present formulas for the Vasserstein distance between two statistical mechanical states of a two-state system. For example, in a ferromagnetic spin-1/2 Ising model the Vasserstein distance is half the difference in the magnetizations.     

Prediction of compositional ordering and separation in alloy nanoclusters

The statistical-mechanical free-energy concentration expansion method (FCEM) in conjunction with semi-empirical coordination-dependent energetic parameters was used for atomistic modeling of Ni-Cu-Al, Ni-Cu-Rh and the corresponding binary Ni-Cu, Ni-Al and Rh-Cu nanocluster systems, containing from 13 to 923 atoms with icosahedral and cuboctahedral shapes. The high efficiency of FCEM enables computations of such relatively large binary or ternary alloy clusters. Remarkable differences, governed by the opposite Ni-Cu and Ni-Al heteroatomic interactions, were noted in the surface segregated “magic-number” ordered compositional patterns of the two ternary clusters. Due to the strong Ni-Al interactions, the compositional ordering extends into the cluster core, and at elevated temperatures a sharp order-disorder transition occurs, depending on the cluster size, shape and composition. The computed site-specific atomic concentrations form the basis for evaluating pertinent thermodynamic functions. For all the alloy clusters a Schottky-type heat capacity anomaly is predicted and attributed to gradual desegregation excitation processes. In addition, inter-cluster compositional separation is computed for Rh-Cu clusters, and transition temperatures estimated from the disappearance of convexity in the free-energy vs. composition curves.