scholarly journals Adiabatic motion and statistical mechanics via mass-zero constrained dynamics

2020 ◽  
Vol 22 (19) ◽  
pp. 10775-10785 ◽  
Author(s):  
Sara Bonella ◽  
Alessandro Coretti ◽  
Rodolphe Vuilleumier ◽  
Giovanni Ciccotti
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Constrained Dynamics ◽  
Numerical Efficiency ◽  
Dft Simulation ◽  
Adiabatic Motion ◽  
Orbital Free

A symplectic, time-reversible algorithm for adiabatically separated systems that exactly samples the Born–Oppenheimer probability distribution is presented and its numerical efficiency is demonstrated on an orbital-free DFT simulation of solid Na.

Many body dynamics

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Statistical Mechanics ◽  
Body Problem ◽  
Many Body ◽  
Complex Objects ◽  
Constrained Dynamics ◽  
Gravitational Systems ◽  
Body Dynamics ◽  
System Temperature ◽  
Speed Up ◽  
Gravitating Systems

Specialized techniques for solving the classical many-body problem are explored in the context of simple gases, more complicated gases, and gravitating systems. The chapter starts with a brief review of some important concepts from statistical mechanics and then introduces the classic Verlet method for obtaining the dynamics of many simple particles. The practical problems of setting the system temperature and measuring observables are discussed. The issues associated with simulating systems of complex objects form the next topic. One approach is to implement constrained dynamics, which can be done elegantly with iterative methods. Gravitational systems are introduced next with stress on techniques that are applicable to systems of different scales and to problems with long range forces. A description of the recursive Barnes-Hut algorithm and particle-mesh methods that speed up force calculations close out the chapter.

scholarly journals Statistical Mechanics of Unconfined Systems: Challenges and Lessons

2021 ◽  
Vol 3 (1) ◽  
pp. 8
Author(s):  
Bruno Arderucio Costa ◽  
Pedro Pessoa
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Maximum Entropy ◽  
A Priori ◽  
Ideal Gas ◽  
Stationary Probability ◽  
De Sitter ◽  
Stationary Probability Distribution ◽  
Local Measurements ◽  
Priori Information

Motivated by applications of statistical mechanics in which the system of interest is spatially unconfined, we present an exact solution to the maximum entropy problem for assigning a stationary probability distribution on the phase space of an unconfined ideal gas in an anti-de Sitter background. Notwithstanding the gas’ freedom to move in an infinite volume, we establish necessary conditions for the stationary probability distribution solving a general maximum entropy problem to be normalizable and obtain the resulting probability for a particular choice of constraints. As a part of our analysis, we develop a novel method for identifying dynamical constraints based on local measurements. With no appeal to a priori information about globally defined conserved quantities, it is therefore applicable to a much wider range of problems.

scholarly journals Matrix theory of correlations in a lattice. Part I

1944 ◽  
Vol 182 (990) ◽  
pp. 244-259 ◽  
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Matrix Theory ◽  
The Other ◽  
Infinite Matrices ◽  
Statistical Correlations ◽  
Unit Cells ◽  
The Mean ◽  
A Chain ◽  
Lattice Part

The statistical mechanics of some crystalline systems may be reduced to statistical correlations between objects which are the unit cells of a fictitious lattice. The correlations are deduced from postulates according to which some configurations of the cells are incompatible with some configurations of the neighbouring cells; if, on the other hand, configurations of neighbours are compatible with each other, their probabilities are to combine by multiplication. By these postulates matrices are implicitly defined such that the probability distribution for a chain of cells is found by forming the powers of a matrix. A similar approach to the statistics of a lattice involves infinite matrices. It does not seem practicable to give explicit expressions for these matrices. If appropriate conditions are complied with, the correlations in a chain are accounted for by adjusting the mean probability coefficients of the cells and for the rest regarding the cells as statistically independent. In this case the infinite matrices may be replaced by the outer power of finite matrices. As result an equation is given by means of which the thermodynamical energy may be calculated as function of temperature.

scholarly journals Necessary and Sufficient Condition for Stability of Generalized Expectation Value

2011 ◽  
Vol 2011 ◽  
pp. 1-7 ◽  
Author(s):  
Aziz El Kaabouchi ◽  
Sumiyoshi Abe
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Complex Systems ◽  
Nonequilibrium Statistical Mechanics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Expectation Value ◽  
Necessary And Sufficient ◽  
Small Deformations ◽  
Generalized Expectation

A class of generalized definitions of expectation value is often employed in nonequilibrium statistical mechanics for complex systems. Here, the necessary and sufficient condition is presented for such a class to be stable under small deformations of a given arbitrary probability distribution.

Random collision processes with transition probabilities belonging to the same type of distribution

1974 ◽  
Vol 11 (4) ◽  
pp. 703-714 ◽  
Author(s):  
Shōichi Nishimura
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Probability Distributions ◽  
Collision Process ◽  
Continuous States ◽  
Collision Processes ◽  
Kac’S Model

By analogy with statistical mechanics we consider a random collision process with discrete time wand continuous states x ∈ [0, ∞). We assume three conditions (i), (ii) and (iii), which can be applied to Kac's model of a Maxwellian gas, and show that the sequence of probability distributions converges to a probability distribution using their moments.

Random collision processes with transition probabilities belonging to the same type of distribution

1974 ◽  
Vol 11 (04) ◽  
pp. 703-714 ◽  
Author(s):  
Shōichi Nishimura
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Probability Distributions ◽  
Collision Process ◽  
Continuous States ◽  
Collision Processes ◽  
Kac’S Model

By analogy with statistical mechanics we consider a random collision process with discrete time wand continuous states x ∈ [0, ∞). We assume three conditions (i), (ii) and (iii), which can be applied to Kac's model of a Maxwellian gas, and show that the sequence of probability distributions converges to a probability distribution using their moments.

scholarly journals Shannon entropy: axiomatic characterization and application

2005 ◽  
Vol 2005 (17) ◽  
pp. 2847-2854 ◽  
Author(s):  
C. G. Chakrabarti ◽  
Indranil Chakrabarty
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Shannon Entropy ◽  
Axiomatic Characterization ◽  
Entropy Function ◽  
Differential Entropy ◽  
Discrete Probability ◽  
Limiting Case ◽  
Modified Entropy ◽  
Classical Entropy

We have presented a new axiomatic derivation of Shannon entropy for a discrete probability distribution on the basis of the postulates of additivity and concavity of the entropy function. We have then modified Shannon entropy to take account of observational uncertainty.The modified entropy reduces, in the limiting case, to the form of Shannon differential entropy. As an application, we have derived the expression for classical entropy of statistical mechanics from the quantized form of the entropy.

scholarly journals Nonextensive Statistical Mechanics: Equivalence Between Dual Entropy and Dual Probabilities

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 594
Author(s):  
George Livadiotis
Keyword(s):  
Probability Distribution ◽  
Statistical Mechanics ◽  
Exponential Distribution ◽  
Internal Energy ◽  
Probability Distributions ◽  
Theoretical Development ◽  
Nonextensive Statistical Mechanics ◽  
Entropy Maximization ◽  
Complex Part

The concept of duality of probability distributions constitutes a fundamental “brick” in the solid framework of nonextensive statistical mechanics—the generalization of Boltzmann–Gibbs statistical mechanics under the consideration of the q-entropy. The probability duality is solving old-standing issues of the theory, e.g., it ascertains the additivity for the internal energy given the additivity in the energy of microstates. However, it is a rather complex part of the theory, and certainly, it cannot be trivially explained along the Gibb’s path of entropy maximization. Recently, it was shown that an alternative picture exists, considering a dual entropy, instead of a dual probability. In particular, the framework of nonextensive statistical mechanics can be equivalently developed using q- and 1/q- entropies. The canonical probability distribution coincides again with the known q-exponential distribution, but without the necessity of the duality of ordinary-escort probabilities. Furthermore, it is shown that the dual entropies, q-entropy and 1/q-entropy, as well as, the 1-entropy, are involved in an identity, useful in theoretical development and applications.

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