scholarly journals PHASE SPACE MEASURE CONCENTRATION FOR AN IDEAL GAS

2009 ◽  
Vol 23 (08) ◽  
pp. 1013-1025 ◽  
Author(s):  
ANTHONY J. CREACO ◽  
NIKOS KALOGEROPOULOS
Keyword(s):  
Phase Space ◽  
Isoperimetric Inequality ◽  
Thermodynamic Limit ◽  
Ideal Gas ◽  
Measure Concentration ◽  
Special Case

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.

scholarly journals ON CANONICAL QUANTIZATION OF THE GAUGED WZW MODEL WITH PERMUTATION BRANES

2011 ◽  
Vol 26 (26) ◽  
pp. 4647-4660
Author(s):  
GOR SARKISSIAN
Keyword(s):  
Riemann Surface ◽  
Phase Space ◽  
Boundary Conditions ◽  
Canonical Quantization ◽  
Simons Theory ◽  
Gauge Fields ◽  
Time Line ◽  
Chern Simons ◽  
Special Case ◽  
Chern Simons Theory

In this paper we perform canonical quantization of the product of the gauged WZW models on a strip with boundary conditions specified by permutation branes. We show that the phase space of the N-fold product of the gauged WZW model G/H on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of the double Chern–Simons theory on a sphere with N holes times the time-line with G and H gauge fields both coupled to two Wilson lines. For the special case of the topological coset G/G we arrive at the conclusion that the phase space of the N-fold product of the topological coset G/G on a strip with boundary conditions given by permutation branes is symplectomorphic to the phase space of Chern–Simons theory on a Riemann surface of the genus N-1 times the time-line with four Wilson lines.

scholarly journals Almost additive entropy

2014 ◽  
Vol 11 (05) ◽  
pp. 1450040 ◽  
Author(s):  
Nikos Kalogeropoulos
Keyword(s):  
Phase Space ◽  
Shannon Entropy ◽  
Power Law ◽  
Physical Significance ◽  
Small Deviations ◽  
Phase Space Volume ◽  
Composition Property ◽  
Flat Manifolds ◽  
Special Case ◽  
Space Volume

We explore consequences of a hyperbolic metric induced by the composition property of the Harvda–Charvat/Daróczy/Cressie–Read/Tsallis entropy. We address the special case of systems described by small deviations of the non-extensive parameter q ≈ 1 from the "ordinary" additive case which is described by the Boltzmann/Gibbs/Shannon entropy. By applying the Gromov/Ruh theorem for almost flat manifolds, we show that such systems have a power-law rate of expansion of their configuration/phase space volume. We explore the possible physical significance of some geometric and topological results of this approach.

scholarly journals Collapsing Cavities and Converging Shocks in Non-Ideal Materials

2019 ◽  
Vol 72 (4) ◽  
pp. 501-520 ◽  
Author(s):  
Zachary M Boyd ◽  
Emma M Schmidt ◽  
Scott D Ramsey ◽  
Roy S Baty
Keyword(s):  
Gas Dynamics ◽  
Nonlinear Eigenvalue Problem ◽  
Ideal Gas ◽  
Test Problems ◽  
Infinite Dimensional ◽  
Cavity Collapse ◽  
Scaling Solution ◽  
Metal Deformation ◽  
Special Case ◽  
The Ideal

Summary As modern hydrodynamic codes increase in sophistication, the availability of realistic test problems becomes increasingly important. In gas dynamics, one common unrealistic aspect of most test problems is the ideal gas assumption, which is unsuited to many real applications, especially those involving high pressure and speed metal deformation. Our work considers the collapsing cavity and converging shock test problems, showing to what extent the ideal gas assumption can be removed from their specification. It is found that while most materials simply do not admit simple (that is scaling) solutions in this context, there are infinite-dimensional families of materials which do admit such solutions. We characterize such materials, derive the appropriate ordinary differential equations and analyze the associated nonlinear eigenvalue problem. It is shown that there is an inherent tension between boundedness of the solution, boundedness of its derivatives and the entropy condition. The special case of a constant-speed cavity collapse is considered and found to be heuristically possible, contrary to common intuition. Finally, we give an example of a concrete non-ideal collapsing cavity scaling solution based on a recently proposed pseudo-Mie–Gruneisen equation of state.

scholarly journals RATE OF PARITY VIOLATION FROM MEASURE CONCENTRATION

2008 ◽  
Vol 23 (03n04) ◽  
pp. 509-517 ◽  
Author(s):  
NIKOS KALOGEROPOULOS
Keyword(s):  
Phase Space ◽  
Statistical Mechanics ◽  
Parity Violation ◽  
Degrees Of Freedom ◽  
Relative Rate ◽  
Measure Concentration ◽  
Space Factor ◽  
Geometric Argument ◽  
Phase Space Factor ◽  
Opposite Chirality

We present a geometric argument determining the kinematic (phase-space) factor contributing to the relative rate at which degrees of freedom of one chirality come to dominate over degrees of freedom of opposite chirality, in models with parity violation. We rely on the measure concentration of a subset of a Euclidean cube which is controlled by an isoperimetric inequality. We provide an interpretation of this result in terms of ideas of statistical mechanics.

ENTROPY AND EXTENDED MEMORY IN DISCRETE CHAOTIC DYNAMICS

1996 ◽  
Vol 06 (04) ◽  
pp. 611-625 ◽  
Author(s):  
WERNER EBELING ◽  
JAN FREUND ◽  
KATJA RATEITSCHAK
Keyword(s):  
Phase Space ◽  
Symbolic Dynamics ◽  
Long Memory ◽  
Chaotic Dynamics ◽  
Logistic Map ◽  
Tent Map ◽  
One Dimensional ◽  
Dynamical Description ◽  
One Dimensional Maps ◽  
Special Case

We investigate simple one-dimensional maps which allow for exact solutions of their related statistical properties. In addition to the originally refined dynamical description a coarsegrained level of description based on certain partitions of the phase space is selected. The deterministic micropscopic dynamics is shifted to a stochastic symbolic dynamics. The higher order entropies are studied for the logistic map, the tent map, and the shark fin map. Markov sources of any prescribed order are constructed explicitly. In a special case, long memory tails are observed. Systems of this type may be of interest for modelling naturally ocurring phenomena.

scholarly journals Entropy minimization, convergence, and Gibbs ensembles (local and global)

2021 ◽  
Author(s):  
◽  
Alexander M. Hughes
Keyword(s):  
Phase Space ◽  
Thermodynamic Limit ◽  
Gibbs Measures ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Canonical Distribution ◽  
Integrable Functions ◽  
Gibbs Entropy ◽  
The Subject ◽  
Grand Canonical

We approach the subject of Statistical Mechanics from two different perspectives. In Part I we adopt the approach of Lanford and Martin-Lof. We examine the minimization of information entropy for measures on the phase space of bounded domains, subject to constraints that are averages of grand canonical distributions. We describe the set of all such constraints and show that it equals the set of averages of all probability measures absolutely continuous with respect to the standard measure on the phase space. We also investigate how the set of constrains relates to the domain of the microcanonical thermodynamic limit entropy. We then show that, for fixed constraints, the parameters of the corresponding grand canonical distribution converge, as volume increases, to the corresponding parameters (derivatives, when they exist) of the thermodynamic limit entropy. In Part II, we use the Banach manifold structure on the space of finite positive measures to show that the critical points of the Gibbs entropy are grand canonical equilibria when the constraints are scalar, and local equilibria when the constraints are integrable functions. This provides a rigorous justification of the derivation of the Gibbs measures that appears often in literature.

scholarly journals On the kinetic method in the new statistics and application in the electron theory of conductivity

1928 ◽  
Vol 119 (783) ◽  
pp. 689-698 ◽  
Keyword(s):  
Phase Space ◽  
Electronic Conductivity ◽  
Kinetic Method ◽  
Dynamical Theory ◽  
Equilibrium States ◽  
Electron Theory ◽  
Classical Physics ◽  
Metallic Conductivity ◽  
New Statistics ◽  
Special Case

Hitherto in work on the new statistics the method of probabilities has always been used, a method which consists in the measuring of the phase space by counting complexions. Now this method is certainly very useful and valuable on account of its generality and convenience, but it has the drawback that only equilibrium states and fluctuations can be treated, not the more general streaming and conductivity phenomena. In the classical physics a method for such problems has been created in the dynamical theory of gases by Max­ well and Boltzmann, and it seems therefore useful to try to adapt it to the new statistics. That task has become particularly necessary, because a corresponding method, due to H. A. Lorentz, has been applied by Sommerfeld* in his theory of metallic conductivity. As it will be shown, the dynamical theory can be worked out in the new statistics, both for the Einstein-Bose and the Fermi-Dirac statistics, in a way completely analogous to the corresponding one of Boltzmann, only some characteristic additional terms occurring. The latter, however, just cancel out in the special case of the electronic conductivity under certain assumptions, so that Sommerfeld’s procedure is justified.

An Isoperimetric Inequality for Convex Polyhedra with Triangular Faces

1968 ◽  
Vol 11 (5) ◽  
pp. 723-727 ◽  
Author(s):  
Magelone Kömhoff
Keyword(s):  
Surface Area ◽  
Isoperimetric Inequality ◽  
Edge Length ◽  
Convex Polyhedra ◽  
Special Case

H. T. Croft [1] has conjectured that among all tetrahedra with fixed total edge length the regular one has the greatest surface area. In this note we prove the following result, which includes this conjecture as a special case

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