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Register a new userThe Virial Correction to the Ideal Gas Law: A Primer
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The virial expansion of a gas is a correction to the ideal gas law that is usually discussed in advanced courses in statistical mechanics. In this note we outline this derivation in a manner suitable for advanced undergraduate and introductory graduate classroom presentations. We introduce a physically meaningful interpretation of the virial expansion that has heretofore escaped attention, by showing that the virial series is actually an expansion in a parameter that is the ratio of the effective volume of a molecule to its mean volume. Using this interpretation we show why under normal conditions ordinary gases such as O_2 and N_2 can be regarded as ideal gases.
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The underlying connection between the degrees of freedom of a system and its nonextensive thermodynamic behavior is addressed. The problem is handled by starting from a thermodynamical system with fractal structure and its analytical reduction to a finite ideal gas. In the limit where the thermofractal has no internal structure, it is found that it reproduces the basic properties of a nonextensive ideal gas with a finite number of particles as recently discussed (Lima & Deppman, Phys. Rev. E 101, 040102(R) 2020). In particular, the entropic $q$-index is calculated in terms of the number of particles both for the nonrelativistic and relativistic cases. In light of such results, the possible nonadditivity or additivity of the entropic structures are also critically analysed and new expressions to the entropy (per particle) for a composed system of thermofractals and its limiting case are derived.
A mathematical procedure is suggested to obtain deformed entropy formulas of type K(S_K) = sum_i P_i K(-ln P_i), by requiring zero mutual K(S_K)-information between a finite subsystem and a finite reservoir. The use of this method is first demonstrated on the ideal gas equation of state with finite constant heat capacity, C, where it delivers the Renyi and Tsallis formulas. A novel interpretation of the qstar = 2-q duality arises from the comparison of canonical subsystem and total microcanonical partition approaches. Finally a new, generalized deformed entropy formula is constructed for the linear relation C(S) = C_0 + C_1 S.
Understanding the physics of glass formation remains one of the major unsolved challenges of condensed matter science. As a material solidifies into a glass, it exhibits a spectacular slowdown of the dynamics upon cooling or compression, but at the same time undergoes only minute structural changes. Among the numerous theories put forward to rationalize this complex behavior, Mode-Coupling Theory (MCT) stands out as the only framework that provides a fully first-principles-based description of glass phenomenology. This review outlines the key physical ingredients of MCT, its predictions, successes, and failures, as well as recent improvements of the theory. We also discuss the extension and application of MCT to the emerging field of non-equilibrium active soft matter
In current experiments with cold quantum gases in periodic potentials, interference fringe contrast is typically the easiest signal in which to look for effects of non-trivial many-body dynamics. In order better to calibrate such measurements, we analyse the background effect of thermal decoherence as it occurs in the absence of dynamical interparticle interactions. We study the effect of optical lattice potentials, as experimentally applied, on the condensed fraction of a non-interacting Bose gas in local thermal equilibrium at finite temperatures. We show that the experimentally observed decrease of the condensate fraction in the presence of the lattice can be attributed, up to a threshold lattice height, purely to ideal gas thermodynamics; conversely we confirm that sharper decreases in first-order coherence observed in stronger lattices are indeed attributable to many-body physics. Our results also suggest that the fringe visibility kinks observed in F.Gerbier et al., Phys. Rev. Lett. 95, 050404 (2005) may be explained in terms of the competition between increasing lattice strength and increasing mean gas density, as the gaussian profile of the red-detuned lattice lasers also increases the effective strength of the harmonic trap.
To illustrate Boltzmanns construction of an entropy function that is defined for a single microstate of a system, we present here the simple example of the free expansion of a one dimensional gas of hard point particles. The construction requires one to define macrostates, corresponding to macroscopic observables. We discuss two different choices, both of which yield the thermodynamic entropy when the gas is in equilibrium. We show that during the free expansion process, both the entropies converge to the equilibrium value at long times. The rate of growth of entropy, for the two choice of macrostates, depends on the coarse graining used to define them, with different limiting behaviour as the coarse graining gets finer. We also find that for only one of the two choices is the entropy a monotonically increasing function of time. Our system is non-ergodic, non-chaotic and essentially non-interacting; our results thus illustrate that these concepts are not very relevant for the question of irreversibility and entropy increase. Rather, the notions of typicality, large numbers and coarse-graining are the important factors. We demonstrate these ideas through extensive simulations as well as analytic results.
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