The classical spin system consisting of three spins with Heisenberg interaction is an example of a completely integrable mechanical system. In this paper we explicitly calculate thermodynamic quantities as density of states, specific heat, susceptibility and spin autocorrelation functions. These calculations are performed (semi-)analytically and shown to agree with corresponding Monte Carlo simulations. For the long-time autocorrelation function, we find, for certain values of the coupling constants, a decay to constant values in the form of an $1/t$ damped harmonic oscillation and propose a theoretical explanation.
The spin-1/2 Hamiltonian for two coupled isosceles Heisenberg triangles, which is well suited for describing the V$_6$-type magnetic molecules, is studied by exact diagonalization. The quantum phase transition diagram, at zero temperature, is obtained as a function of the theoretical parameters. The zero temperature magnetization is also obtained as a function of the external magnetic field. The thermodynamic behavior of the magnetization, entropy, susceptibility, and specific heat, as a function of temperature, are also computed and the corresponding magnetocaloric effect analyzed for various…
The triangle distribution function f^(3) for three mutual nearest neighbors in the plane describes basic aspects of short-range order and statistical thermodynamics in two-dimensional many-particle systems. This paper examines prospects for constructing a self-consistent calculation for the rigid-disk system f^(3). We present several identities obeyed by f^(3). A rudimentary closure suggested by scaled-particle theory is introduced. In conjunction with three of the basic identities, this closure leads to a unique f^(3) over the entire density range. The pressure equation of state exhibits qual…
Ground state and thermodynamics of geometrically frustrated spin-1/2 Ising-Heisenberg model on two different but topologically related triangles-in-triangles lattices is investigated in particular. A rigorous mapping based on generalized star-triangle transformations enables one to derive exact analytical results for the free and internal energy, spontaneous magnetization, entropy and specific heat from the known exact results for the corresponding spin-1/2 Ising model on a triangular lattice. It is demonstrated that several unconventional quantum ordered and disordered phases may become the g…
At arbitrary temperature $T$, we solve for the dynamics of single molecule magnets composed of three classical Heisenberg spins either on a chain with two equal exchange constants $J_1$, or on an isosceles triangle with a third, different exchange constant $J_2$. As $T\rightrarrow\infty$, the Fourier transforms and long-time asymptotic behaviors of the two-spin time correlation functions are evaluated exactly. The lack of translational symmetry on a chain or an isosceles triangle yields time correlation functions that differ strikingly from those on an equilateral trinagle with $J_1=J_2$. At l…
We study the thermodynamics of classical Heisenberg model using the multipath approach to Metropolis algorithm Monte Carlo simulation. This simulation approach produces uncorrelated results with known precision. Also, it can be easily generalized to other classical models of magnetism. Comparing results obtained from multipath and from single--path simulations we demonstrate that these approaches produce equivalent results.
We investigate the thermodynamics of integrable classical field theories under the effect of a random initial configuration, motivated by the nonequilibrium evolution of quantum field theories. The approach to thermal equilibrium is analysed with numerical simulations and confronted with analytical results. The possibility of comparing the classical results with their quantum counterparts could be a way of inferring the behaviour of out-of-equilibrium quantum field theories during the thermalization process.
Thermodynamics of itinerant magnets is studied using a classical model with one parameter characterizing the degree of itinerancy. Monte Carlo simulations for bcc and fcc lattices are compared with the mean-field approximation and with the Onsager cavity field approximation extended to itinerant systems. The qualitative features of thermodynamics are similar to the known results of the functional integral method. It is found that magnetic short-range order is weak and almost independent on the degree of itinerancy, and the mean-field approximation describes the thermodynamics reasonably well. …
The thermodynamic properties of a classical d-dimensional spin-S Heisenberg ferromagnet, with long-range interactions decaying as $r^{-p}$ and in the presence of an external magnetic field, is investigated by means of the spectral density method in the framework of classical statistical mechanics. We find that long-range order exists at finite temperature for $d<p<2d$ with $d\leq 2$ and for $p>d$ with $d>2$, consistently with known theorems. Besides, the related critical temperature is determined and a study of the critical properties is performed.
We study the statistical and dynamical properties of the quantum triangle map, whose classical counterpart can exhibit ergodic and mixing dynamics, but is never chaotic. Numerical results show that ergodicity is a sufficient condition for spectrum and eigenfunctions to follow the prediction of Random Matrix Theory, even though the underlying classical dynamics is not chaotic. On the other hand, dynamical quantities such as the out-of-time-ordered correlator (OTOC) and the number of harmonics, exhibit a growth rate vanishing in the semiclassical limit, in agreement with the fact that classical …
The total entropy production quantifies the extent of irreversibility in thermodynamic systems, which is nonnegative for any feasible dynamics. When additional information such as the initial and final states or moments of an observable is available, it is known that tighter lower bounds on the entropy production exist according to the classical speed limits and the thermodynamic uncertainty relations. Here, we obtain a universal lower bound on the total entropy production in terms of probability distributions of an observable in the time forward and backward processes. For a particular case, …
The exact statistical-mechanical solution for the equilibrium properties, both thermodynamic and structural, of one-dimensional fluids of particles interacting via the triangle-well and the ramp potentials is worked out. In contrast to previous studies, where the radial distribution function $g(r)$ was obtained numerically from the structure factor by Fourier inversion, we provide a fully analytic representation of $g(r)$ up to any desired distance. The solution is employed to perform an extensive study of the equation of state, the excess internal energy per particle, the residual multipartic…
A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to fill this gap. We describe some aspects of these arrays, which generalize similar properties of the binomial coefficients. Some combinatorial models enumerated by polynomial coefficients, including lattice paths model, spin chain model and scores in a drawing game, are introduced. Several known binomial identities are then extended. In addition, we calculat…
A mixed Ising-Heisenberg spin system consisting of triangular XXZ-Heisenberg spin clusters assembled into a chain by alternating with Ising spins interacting to all three spins in the triangle is considered. The exact solution of the model is given in terms of the generalized decoration--iteration map and within the transfer-matrix technique. Exact expressions for thermodynamic functions are derived. Ground state phase diagrams, thermodynamic and magnetic properties of the system are examined.
Low-temperature thermodynamics of the classical frustrated ferromagnetic spin chain is studied. Using transfer-matrix method we found the behavior of the correlation function and zero-field susceptibility at the ferromagnetic-helical transition point. It is shown that the critical exponent for the susceptibility is changed from 2 to 4/3 at the transition point.
We study the antiferromagnetic classical Ising (AFI) model on the sorrel net, a 1/9th site depleted and 1/7th bond depleted triangular lattice. Our classical Monte Carlo simulations, verified by exact results for small system sizes, show that the AFI model on this corner-shared triangle net (with coupling constant $J_1$) is highly frustrated, with a residual entropy of $\frac{S}{N}$= 0.48185$\pm$0.00008. Anticipating that it may be difficult to achieve perfect bond depletion, we investigate the physics originating from turning back on the deleted bonds ($J_2$) to create a lattice of edge-shari…
The geometric frustration of the spin-1/2 Ising-Heisenberg model on the triangulated Kagome (triangles-in-triangles) lattice is investigated within the framework of an exact analytical method based on the generalized star-triangle mapping transformation. Ground-state and finite-temperature phase diagrams are obtained along with other exact results for the partition function, Helmholtz free energy, internal energy, entropy, and specific heat, by establishing a precise mapping relationship to the corresponding spin-1/2 Ising model on the Kagome lattice. It is shown that the residual entropy of t…
We investigate perturbative thermodynamic geometry of nonextensive ideal Classical, Bose and Fermi gases.We show that the intrinsic statistical interaction of nonextensive Bose (Fermi) gas is attractive (repulsive) similar to the extensive case but the value of thermodynamic curvature is changed by nonextensive parameter. In contrary to the extensive ideal classical gas, the nonextensive one may be divided to two different regimes. According to deviation parameter of the system to the nonextensive case, one can find a special value of fugacity, $z^{*}$, where the sign of thermodynamic curvatur…
We study the formulation of statistical mechanics on noncommutative classical phase space, and construct the corresponding canonical ensemble theory. For illustration, some basic and important examples are considered in the framework of noncommutative statistical mechanics: such as the ideal gas, the extreme relativistic gas, and the 3-dimensional harmonic oscillator.
The basic idea of a microscopic understanding of Thermodynamics is to derive its main features from a microscopic probability distribution. In such a vein, we investigate the thermal statistics of quasi-probabilities's semi-classical analogs in phase space for the important case of quadratic Hamiltonians, focusing attention in the three more important instances, i.e., those of Wigner, $P$-, and Husimi distributions. Introduction of an effective temperature permits one to obtain a unified thermodynamic description that encompasses and unifies the three different quasi-probability distributions.…
We present an analysis of the foundations of the well known Clausius inequality. It is shown that, strictly speaking, the inequality is not a logical consequence of the Kelvin-Planck formulation of the second law of thermodynamics. Some thought experiments demonstrating the violation of the Clausius inequality are considered. Also, a reformulation of the Landauer's principle in terms of the Clausius inequality is proposed. This version of the inequality may be considered a consequence of the fluctuation theorem.
This work presents an efficient numerical method to evaluate the free energy density and associated thermodynamic quantities of (quasi) one-dimensional classical systems, by combining the transfer operator approach with a numerical discretization of integral kernels using quadrature rules. For analytic kernels, the technique exhibits exponential convergence in the number of quadrature points. As demonstration, we apply the method to a classical particle chain, to the semiclassical nonlinear Schrödinger equation and to a classical system on a cylindrical lattice.
We analyze an engine whose working fluid consists of a single quantum particle, paralleling Szilard's construction of a classical single-particle engine. Following his resolution of Maxwell's Second Law paradox using the latter, which turned on physically instantiating the demon (control subsystem), the quantum engine's design mirrors the classically-chaotic Szilard Map that operates a thermodynamic cycle of measurement, thermal-energy extraction, and memory reset. Focusing on the thermodynamic costs to observe and control the particle and comparing these in the quantum and classical limits, w…
We study the configurational probability distribution of a mono-atomic gas with a finite number of particles N in the micro-canonical ensemble. We give two arguments why the thermodynamic entropy of the configurational subsystem involves Renyi's entropy function rather than that of Tsallis. The first argument is that the temperature of the configurational subsystem is equal to that of the kinetic subsystem. The second argument is that the instability of the pendulum, which occurs for energies close to the rotation threshold, is correctly reproduced.
Heat can flow from cold to hot at any phase separation. Therefore Lynden-Bell's gravo-thermal catastrophe must be reconsidered. The original objects of Thermodynamics, the separation of phases at first order phase transitions, like boiling water in steam engines, are not described by a single canonical ensemble. Inter-phase fluctuations are not covered. The basic principles of statistical mechanics, especially of phase transitions have to be reconsidered without the use of the thermodynamic limit. Then thermo-statistics applies also to nuclei and large astronomical systems. A lot of similarity…
