Macroscopic equations for the adiabatic piston

A simplified version of a classical problem in thermodynamics -- the adiabatic piston -- is discussed in the framework of kinetic theory. We consider the limit of gases whose relaxation time is extremely fast so that the gases contained on the left and right chambers of the piston are always in equilibrium (that is the molecules are uniformly distributed and their velocities obey the Maxwell-Boltzmann distribution) after any collision with the piston. Then by using kinetic theory we derive the collision statistics from which we obtain a set of ordinary differential equations for the evolution of the macroscopic observables (namely the piston average velocity and position, the velocity variance and the temperatures of the two compartments). The dynamics of these equations is compared with simulations of an ideal gas and a microscopic model of gas settled to verify the assumptions used in the derivation. We show that the equations predict an evolution for the macroscopic variables which catches the basic features of the problem. The results here presented recover those derived, using a different approach, by Gruber, Pache and Lesne in J. Stat. Phys. 108, 669 (2002) and 112, 1177 (2003).Comment: 13 pages, 7 figures (revTeX4) The paper has been completely rewritten with new derivation and results, supplementary information can be found at http://denali.phys.uniroma1.it/~cencini/Papers/cppv07_supplements.pd

Quantum thermodynamics with missing reference frames: Decompositions of free energy into non-increasing components

If an absolute reference frame with respect to time, position, or orientation is missing one can only implement quantum operations which are covariant with respect to the corresponding unitary symmetry group G. Extending observations of Vaccaro et al., I argue that the free energy of a quantum system with G-invariant Hamiltonian then splits up into the Holevo information of the orbit of the state under the action of G and the free energy of its orbit average. These two kinds of free energy cannot be converted into each other. The first component is subadditive and the second superadditive; in the limit of infinitely many copies only the usual free energy matters. Refined splittings of free energy into more than two independent (non-increasing) terms can be defined by averaging over probability measures on G that differ from the Haar measure. Even in the presence of a reference frame, these results provide lower bounds on the amount of free energy that is lost after applying a covariant channel. If the channel properly decreases one of these quantities, it decreases the free energy necessarily at least by the same amount, since it is unable to convert the different forms of free energies into each other.Comment: 17 pages, latex, 1 figur

Thermodynamic curvature measures interactions

Thermodynamic fluctuation theory originated with Einstein who inverted the relation $S=k_B\ln\Omega$ to express the number of states in terms of entropy: $\Omega= \exp(S/k_B)$. The theory's Gaussian approximation is discussed in most statistical mechanics texts. I review work showing how to go beyond the Gaussian approximation by adding covariance, conservation, and consistency. This generalization leads to a fundamentally new object: the thermodynamic Riemannian curvature scalar $R$, a thermodynamic invariant. I argue that $|R|$ is related to the correlation length and suggest that the sign of $R$ corresponds to whether the interparticle interactions are effectively attractive or repulsive.Comment: 29 pages, 7 figures (added reference 27

Microcanonical quantum fluctuation theorems

Previously derived expressions for the characteristic function of work performed on a quantum system by a classical external force are generalized to arbitrary initial states of the considered system and to Hamiltonians with degenerate spectra. In the particular case of microcanonical initial states explicit expressions for the characteristic function and the corresponding probability density of work are formulated. Their classical limit as well as their relations to the respective canonical expressions are discussed. A fluctuation theorem is derived that expresses the ratio of probabilities of work for a process and its time reversal to the ratio of densities of states of the microcanonical equilibrium systems with corresponding initial and final Hamiltonians.From this Crooks-type fluctuation theorem a relation between entropies of different systems can be derived which does not involve the time reversed process. This entropy-from-work theorem provides an experimentally accessible way to measure entropies.Comment: revised and extended versio

Internal thermal noise in the LIGO test masses : a direct approach

The internal thermal noise in LIGO's test masses is analyzed by a new technique, a direct application of the Fluctuation-Dissipation Theorem to LIGO's readout observable, $x(t)=$(longitudinal position of test-mass face, weighted by laser beam's Gaussian profile). Previous analyses, which relied on a normal-mode decomposition of the test-mass motion, were valid only if the dissipation is uniformally distributed over the test-mass interior, and they converged reliably to a final answer only when the beam size was a non-negligible fraction of the test-mass cross section. This paper's direct analysis, by contrast, can handle inhomogeneous dissipation and arbitrary beam sizes. In the domain of validity of the previous analysis, the two methods give the same answer for $S_x(f)$, the spectral density of thermal noise, to within expected accuracy. The new analysis predicts that thermal noise due to dissipation concentrated in the test mass's front face (e.g. due to mirror coating) scales as $1/r_0^2$, by contrast with homogeneous dissipation, which scales as $1/r_0$ ($r_0$ is the beam radius); so surface dissipation could become significant for small beam sizes.Comment: 6 pages, RevTex, 1 figur

Ignorance based inference of optimality in thermodynamic processes

We derive ignorance based prior distribution to quantify incomplete information and show its use to estimate the optimal work characteristics of a heat engine.Comment: Latex, 10 pages, 3 figure

Isotropic, Nematic and Smectic A Phase Behaviour in a Fictitious Field

Phase behaviours of liquid crystals under external fields, conjugate to the nematic order and smectic order, are studied within the framework of mean field approximation developed by McMillan. It is found that phase diagrams, of temperature vs interaction parameter of smectic A order, show several topologically different types caused by the external fields. The influences of the field conjugate to the smectic A phase, which is fictitious field, are precisely discussed.Comment: To be published in J. Phys. Soc. Jpn. vol.73 No.

Entropy of Extremal Black Holes in Two Dimensions

Entropy for two dimensional extremal black holes is computed explicitly in a finite-space formulation of the black hole thermodynamics and is shown to be zero {\it locally}. Our results are in conformity with the recent one by Hawking et al in four dimensions.Comment: 11 page

Calculation of the Chiral Lagrangian Coefficients

We present a systematic way to combine the global color model and the instanton liquid model to calculate the chiral Lagrangian coefficients. Our numerical results are in agreement well with the experimental values.Comment: 7 pages, To appear in Chin.Phys.Lett, Year 200

Dissipative Particle Dynamics with Energy Conservation

The stochastic differential equations for a model of dissipative particle dynamics with both total energy and total momentum conservation in the particle-particle interactions are presented. The corresponding Fokker-Planck equation for the evolution of the probability distribution for the system is deduced together with the corresponding fluctuation-dissipation theorems ensuring that the ab initio chosen equilibrium probability distribution for the relevant variables is a stationary solution. When energy conservation is included, the system can sustain temperature gradients and heat flow can be modeled.Comment: 7 pages, submitted to Europhys. Let