Narrow Results

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.

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.

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.