Narrow Results

In this paper, using a deformed algebra $[X,P]=i\hslash /(1-{\alpha }^2{P}^2)$ which is originated from various theories of gravity, we study thermodynamical properties of the classical and extreme relativistic gases in canonical ensembles. In this regards, we exactly calculate the modified partition function, Helmholtz free energy, internal energy, entropy, heat capacity and the thermal pressure which conclude to the familiar form of the equation of state for the ideal gas. The advantage of applying this algebra is not only considering all natural cutoffs but also its structure is similar to the other effective quantum gravity models such as polymer, Snyder and noncommutative space–time frameworks. Moreover, after obtaining some thermodynamical quantities including internal energy and entropy, we conclude at high temperature limits due to the decreasing of the number of microstates, these quantities reach to maximal bounds which do not exist in standard cases and it concludes that at the presence of gravity for both micro-canonic and canonic ensembles, the internal energy and the entropy tend to these upper bounds.

We have studied the thermodynamic properties of noninteracting gases in periodic lattice potential at arbitrary integer fillings and compared them with that of ideal homogeneous gases. By deriving explicit expressions for the thermodynamic quantities and performing exact numerical calculations, we have found that the dependence of e.g., entropy and energy on the temperature in the normal phase is rather weak especially at large filling factors. In the Bose condensed phase, their power dependence on the reduced temperature is nearly linear, which is in contrast to that of ideal homogeneous gases. We evaluated the discontinuity in the slope of the specific heat which turned out to be approximately the same as that of the ideal homogeneous Bose (IHB) gas for filling factor ν = 1. The discontinuity i.e. the jump in the heat capacity per particle linearly decreases with increasing ν. These results may serve as a checkpoint for various experiments on optical lattices as well as theoretical studies of weakly interacting Bose systems in periodic potentials being a starting point for perturbative calculations.

We point out that a special case of an ideal gas exhibits concentration of the volume of its phase space, which is a sphere, around its equator in the thermodynamic limit. The rate of approach to the thermodynamic limit is determined. Our argument relies on the spherical isoperimetric inequality of Lévy and Gromov.

In confined space, the thermodynamic potential is shape-dependent. Therefore, the pressure of ideal gases in confined space is anisotropic. We study this anisotropy in a thermodynamic manner and find that the thermodynamic pressures usually depend on the form of deformations, and hence are not equal to each other which is a natural representation of the anisotropic mechanical properties of a confined ideal gas. We also find that the boundary effects are much more significant than the statistical fluctuations under low-temperature and high-density conditions. Finally, we show that there is little difference between the boundary effects in 2D space and those in 3D space.

Classical statistical thermodynamics was developed in the second half of the 19th century. The implied fact is that since then all processes and effects that fall into the domain of classical statistical thermodynamics have been discovered and explained. However, analyzing the outputs from our numerical model of an ideal gas, we noticed that the atoms separate during the expansion of the gas into empty space. This separation occurs as a consequence of the distribution of molecules by velocities. Namely, faster molecules will move further away from their initial position than slower ones. The objective of this paper is to describe the existence of this process and to present some basic physical consequences of this effect.

The Bousso entropy bound is investigated for static, spherically symmetric configurations of an ideal gas with a Bose–Einstein or Fermi–Dirac distribution function. Gas of massive particles is considered. This paper is continuation of the previous work concerning the massless case. Special attention is devoted to light sheets generated by spheres. Conditions under which the Bousso bound can be violated are discussed and it is shown that a possible violating region cannot be arbitrarily large and that it is contained inside a sphere of unit Planck radius if the number of independent spin states g_s is small enough. It is also shown that the central temperature must exceed the Planck temperature in order to get a violation of the Bousso bound for g_s which is not too large. The situation for higher-dimensional space–times is also discussed and the conditions of Flanagan, Marolf and Wald are investigated.

We critically analyze the models of a main-sequence star based on polytropic gases. First, we put in evidence that the hypothesis of polytropic gas is compatible with the constitutive equation of an ideal gas if the transformations inside the star are adiabatic. Then, neither the mono- or the polyphasic models of an ideal gas can be applied inside the stellar core for the existence in this region of nuclear reactions. Moreover, we prove that polyphasic models doesn’t allow the existence of C¹ solutions of the stationary hydrodynamic equations.

The physical basis of the canonical and grand canonical distributions is questioned. In particular, we question the usual methods by which these distributions are derived, namely that fluctuations in entropy around energy and particle number are assumed to occur when the entropy depends only on variables which cannot themselves fluctuate. We show, starting from the Maxwellian velocity distribution, that the probability that a classical ideal gas at a fixed temperature occupies a given energy state corresponds not to the canonical ensemble of classical statistical mechanics but to the Gamma distribution. Computer simulations of a hard-sphere fluid demonstrate the principles. The analysis is extended to open systems in which the number of particles fluctuates and we show that for a system connected to a particle reservoir the Poisson distribution governs the probability of finding a given number of particles. The resulting probability distributions are used to calculate the Shannon information entropy which is then compared with the thermodynamic entropy. We argue that information theoretic entropy and thermodynamic entropy, whilst related, are not necessarily identical and that the information entropy contains non-thermodynamic components.

Clausius’ original papers covering the re-working of Carnot’s theory and the subsequent development of the concept of the entropy of a body are reviewed critically. We show that Clausius’ thinking was dominated by the then prevalent idea that a body contained heat and argue that his concept of aequivalenzwerth, the forerunner of entropy, should be understood as the quantity that measures the ability of the heat in a body to be transformed into work. Although this view of heat had been rejected by the mid-1870s, the concept of the entropy of a body not only survived but went on to dominate thinking in thermodynamics. We draw attention to several contradictions within Clausius’ papers, among them the idea that only infinitesimal quasi-static processes are reversible even though reversibility is integral to the theory of motive power. In addition, Clausius developed his theory of entropy to account for internal work done in overcoming the forces between particles but then applied the theory to the ideal gas in which there are no such forces. We show how this conflicts with the First Law and discuss whether entropy is the state function Clausius claimed it to be.