Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDuality Symmetry, Two Entropy Functions, and an Eigenvalue Problem in Gibbs Theory
74
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We generalize the convex duality symmetry in Gibbs statistical ensemble formulation, between Massieus free entropy $Phi_{V,N} (beta)$ and the Gibbs entropy $varphi_{V,N}(u)$ as a function of mean internal energy $u$. The duality tells us that Gibbs thermodynamic entropy is to the law of large numbers (LLN) for arithmetic sample means what Shannons information entropy is to the LLN for empirical counting frequencies. Following the same logic, we identify $u$ as the conjugate variable to counting frequency, a Hamilton-Jacobi equation for Shannon entropy as an equation of state, and suggest an eigenvalue problem for modeling statistical frequencies of correlated data.
rate research
Read More
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
This paper introduces the basic concepts of information theory. Based on these concepts, we regard the states in the state space and the types of ideal gases as the symbols in a symbol set to calculate the mixing entropy of ideal gas involved in Gibbs Paradox. The discussion above reveals that the non-need for distinguishing can resolve the contradiction of Gibbs Paradox, implying the introduction of indistinguishability is not necessary. Further analysis shows that the information entropy of gas molecular types does not directly correlate to the energy of a gas system, so it should not be used for calculating thermodynamic and statistical dynamic entropies. Therefore, the mixing entropy of the ideal gas is independent of the molecular types and is much smaller than the value commonly thought.
We show that the exact beta-function beta(g) in the continuous 2D gPhi^{4} model possesses the Kramers-Wannier duality symmetry. The duality symmetry transformation tilde{g}=d(g) such that beta(d(g))=d(g)beta(g) is constructed and the approximate values of g^{*} computed from the duality equation d(g^{*})=g^{*} are shown to agree with the available numerical results. The calculation of the beta-function beta(g) for the 2D scalar gPhi^{4} field theory based on the strong coupling expansion is developed and the expansion of beta(g) in powers of g^{-1} is obtained up to order g^{-8}. The numerical values calculated for the renormalized coupling constant g_{+}^{*} are in reasonable good agreement with the best modern estimates recently obtained from the high-temperature series expansion and with those known from the perturbative four-loop renormalization-group calculations. The application of Cardys theorem for calculating the renormalized isothermal coupling constant g_{c} of the 2D Ising model and the related universal critical amplitudes is also discussed.
We discuss the implementation of two different truncated Generalized Gibbs Ensembles (GGE) describing the stationary state after a mass quench process in the Ising Field Theory. One truncated GGE is based on the semi-local charges of the model, the other on regulariz
We derive exact analytic results for several four-point correlation functions for statistical models exhibiting phase separation in two-dimensions. Our theoretical results are then specialized to the Ising model on the two-dimensional strip and found to be in excellent agreement with high-precision Monte Carlo simulations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
