scholarly journals Thermodynamics Beyond Molecules: Statistical Thermodynamics of Probability Distributions

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 890 ◽  
Author(s):  
Themis Matsoukas
Keyword(s):  
Probability Distributions ◽  
Variational Calculus ◽  
Statistical Thermodynamics ◽  
Fundamental Question ◽  
Physical Models ◽  
Probabilistic Argument ◽  
Molecular Systems ◽  
Space Of Distributions ◽  
Thermodynamic Relationship ◽  
Special Case

Statistical thermodynamics has a universal appeal that extends beyond molecular systems, and yet, as its tools are being transplanted to fields outside physics, the fundamental question, what is thermodynamics, has remained unanswered. We answer this question here. Generalized statistical thermodynamics is a variational calculus of probability distributions. It is independent of physical hypotheses but provides the means to incorporate our knowledge, assumptions and physical models about a stochastic processes that gives rise to the probability in question. We derive the familiar calculus of thermodynamics via a probabilistic argument that makes no reference to physics. At the heart of the theory is a space of distributions and a special functional that assigns probabilities to this space. The maximization of this functional generates the mathematical network of thermodynamic relationship. We obtain statistical mechanics as a special case and make contact with Information Theory and Bayesian inference.

scholarly journals The extended odd family of probability distributions with practice to a submodel

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3855-3867 ◽  
Author(s):  
Hassan Bakouch ◽  
Christophe Chesneau ◽  
Muhammad Khan
Keyword(s):  
Heavy Tails ◽  
Maximum Likelihood Method ◽  
Probability Distributions ◽  
Likelihood Method ◽  
Tuning Parameter ◽  
Data Sets ◽  
New Family ◽  
Rate Functions ◽  
Special Case ◽  
Family Of Distributions

In this paper, we introduce a new family of distributions extending the odd family of distributions. A new tuning parameter is introduced, with some connections to the well-known transmuted transformation. Some mathematical results are obtained, including moments, generating function and order statistics. Then, we study a special case dealing with the standard loglogistic distribution and the modifiedWeibull distribution. Its main features are to have densities with flexible shapes where skewness, kurtosis, heavy tails and modality can be observed, and increasing-decreasing-increasing, unimodal and bathtub shaped hazard rate functions. Estimation of the related parameters is investigated by the maximum likelihood method. We illustrate the usefulness of our extended odd family of distributions with applications to two practical data sets.

Equilibrium probability distributions for low density highway traffic

1966 ◽  
Vol 3 (1) ◽  
pp. 247-260 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Poisson Process ◽  
Probability Distributions ◽  
Initial Distribution ◽  
Order Effects ◽  
Highway Traffic ◽  
Wide Range ◽  
Headway Distribution ◽  
Equilibrium Distributions ◽  
Initial Distributions ◽  
Special Case

If on a long homogeneous highway there is no interaction between cars, then, under a wide range of conditions, an initial distribution of cars will in the course of time tend toward that of a Poisson process with statistically independent velocities for the cars in any finite interval of highway. Here we will generalize this known property to obtain the following. Suppose cars do interact in such a way as to delay a car when it passes another, but the density of cars is so low that we can neglect simultaneous interactions between three or more cars. There will again be equilibrium distributions of cars to which general classes of initial distributions will converge. These equilibrium distributions are superpositions of two statistically independent processes, one a Poisson process of single free cars with statistically independent velocities, and the other a Poisson process of interacting pairs of cars with various velocities. In the limit of zero interaction, the density of pairs vanishes leaving only the Poisson process of single cars as a special case. To the same order of approximation, including the first order effects of interactions, the headway distribution between consecutive cars will still have exponential tail outside the range of interaction.

scholarly journals A note on Lévy’s Brownian motion II

1989 ◽  
Vol 114 ◽  
pp. 165-172 ◽  
Author(s):  
Si Si
Keyword(s):  
Brownian Motion ◽  
Random Fields ◽  
Probability Distributions ◽  
Joint Probability ◽  
Variational Calculus ◽  
Normal Derivative ◽  
Gaussian Random Fields ◽  
Normal Direction ◽  
Joint Probability Distributions ◽  
Derivatives Of

The purpose of this paper is to discuss some particular random fields derived from Lévy’s Brownian motion to find its characteristic properties of the joint probability distributions. In [9], special attention was paid to the behaviour of the Brownian motion when the parameter runs along a curve in the parameter space, and with this property the conditional expectation has been obtained when the values are known on the curve.The present paper deals with the variation of the Brownian motion in the normal direction to a given curve, in contrast to the case in [9], where we discussed the properties along the curve. Actually we shall find, in this paper, formulae of the variation with the help of the normal derivative of Brownian motion and observe its singularity. We then discuss partial derivatives of Rd-parameter Lévy’s Brownian motion and make attempt to restrict the parameter to a hypersurface so that we obtain new random fields on that hypersurface. By comparing such derivatives with those of other Gaussian random fields, we can see that the singularity of the new random fields seems to be an interesting characteristic of Lévy’s Brownian motion. Further, we hope that our approach may be thought of as a first step to the variational calculus for Gaussian random fields.

Optimum Profile of Thin Fins With Volumetric Heat Generation: A Unified Approach

2005 ◽  
Vol 127 (8) ◽  
pp. 945-948 ◽  
Author(s):  
B. Kundu ◽  
P. K. Das
Keyword(s):  
Closed Form ◽  
Heat Generation ◽  
Variational Calculus ◽  
Optimality Criteria ◽  
Closed Form Expression ◽  
Pin Fins ◽  
Fin Thickness ◽  
Optimum Profile ◽  
Special Case ◽  
Volumetric Heat Generation

In this paper, a generalized methodology for the optimum design of thin fins with uniform volumetric heat generation is described. Using variational calculus, the optimum profiles of longitudinal, annular, and pin fins are determined from the basic fin equation under the constraint of specified fin volume. From a common optimality criteria, a generalized closed form expression for the fin thickness is obtained for the above three types of fins. Closed-form expressions are also obtained for the optimum profiles of longitudinal and pin fins. As a special case, both the temperature profile and the shape of optimum fins without heat generation are determined.

scholarly journals Constrained Nets for Graph Matching and Other Quadratic Assignment Problems

1991 ◽  
Vol 3 (2) ◽  
pp. 268-281 ◽  
Author(s):  
Petar D. Simić
Keyword(s):  
Graph Matching ◽  
Optimization Problems ◽  
Quadratic Assignment Problem ◽  
Semiclassical Approximation ◽  
Geometric Optimization ◽  
Elastic Net ◽  
Physical Models ◽  
Quadratic Assignment ◽  
Matching Problem ◽  
Special Case

Some time ago Durbin and Willshaw proposed an interesting parallel algorithm (the “elastic net”) for approximately solving some geometric optimization problems, such as the Traveling Salesman Problem. Recently it has been shown that their algorithm is related to neural networks of Hopfield and Tank, and that they both can be understood as the semiclassical approximation to statistical mechanics of related physical models. The main point of the elastic net algorithm is seen to be in the way one deals with the constraints when evaluating the effective cost function (free energy in the thermodynamic analogy), and not in its geometric foundation emphasized originally by Durbin and Willshaw. As a consequence, the elastic net algorithm is a special case of the more general physically based computations and can be generalized to a large class of nongeometric problems. In this paper we further elaborate on this observation, and generalize the elastic net to the quadratic assignment problem. We work out in detail its special case, the graph matching problem, because it is an important problem with many applications in computational vision and neural modeling. Simulation results on random graphs, and on structured (hand-designed) graphs of moderate size (20-100 nodes) are discussed.

Learning the Gestalt Rule of Collinearity from Object Motion

2003 ◽  
Vol 15 (8) ◽  
pp. 1865-1896 ◽  
Author(s):  
Carsten Prodöhl ◽  
Rolf P. Würtz ◽  
Christoph von der Malsburg
Keyword(s):  
Visual Experience ◽  
Moving Objects ◽  
A Priori ◽  
Object Motion ◽  
Fundamental Question ◽  
Video Sequences ◽  
Transient Responses ◽  
Self Organized ◽  
The Brain ◽  
Special Case

The Gestalt principle of collinearity (and curvilinearity) is widely regarded as being mediated by the long-range connection structure in primary visual cortex. We review the neurophysiological and psychophysical literature to argue that these connections are developed from visual experience after birth, relying on coherent object motion. We then present a neural network model that learns these connections in an unsupervised Hebbian fashion with input from real camera sequences. The model uses spatiotemporal retinal filtering, which is very sensitive to changes in the visual input. We show that it is crucial for successful learning to use the correlation of the transient responses instead of the sustained ones. As a consequence, learning works best with video sequences of moving objects. The model addresses a special case of the fundamental question of what represents the necessary a priori knowledge the brain is equipped with at birth so that the self-organized process of structuring by experience can be successful.

Choice under uncertainty

2020 ◽  
pp. 71-131
Author(s):  
David M. Kreps
Keyword(s):  
Utility Function ◽  
Expected Utility ◽  
Probability Distributions ◽  
Market Demand ◽  
Von Neumann ◽  
Consumption Decisions ◽  
Choice Under Uncertainty ◽  
States Of Nature ◽  
Special Case

This chapter examines how many important consumption decisions concern choices, the consequences of which are uncertain at the time the choice is made. It begins with the theory of von Neumann–Morgenstern expected utility. In this theory, uncertain prospects are modeled as probability distributions over a given set of prizes. That is, the probabilities of various prizes are given as part of the description of the object. The chapter then takes up the special case where the prizes are amounts of money; then one is able to say a bit more about the nature of the utility function that represents preferences. It discusses a few applications of this theory to the topic of market demand. Finally, the chapter turns to a richer theory, where uncertain prospects are functions from “states of nature” to prizes, and where probabilities arise subjectively, as part of the representation of a consumer's preferences.

scholarly journals Method to Remove Tilt-to-Length Coupling Caused by Interference of Flat-Top Beam and Gaussian Beam

2019 ◽  
Vol 9 (19) ◽  
pp. 4112 ◽  
Author(s):  
Ya Zhao ◽  
Zhi Wang ◽  
Yupeng Li ◽  
Chao Fang ◽  
Heshan Liu ◽  
...  
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Gaussian Beam ◽  
Laser Interferometer ◽  
Physical Models ◽  
Space Antenna ◽  
Coupling Noise ◽  
Laser Interferometer Space Antenna ◽  
Flat Top Beam ◽  
Special Case

We discuss the tilt-to-length (TTL) coupling noise caused by interference between a flat-top beam and a Gaussian beam. Several physical models are presented to research the effects of non-diffracted and diffracted beams on TTL noise. A special case that can remove TTL coupling noise is discovered and is verified via both theoretical analysis and numerical simulations. The proposed case could provide desirable suggestions for the construction of high-precision interferometers such as the Laser Interferometer Space Antenna (LISA), Taiji program, or other interferometry systems.

Properties of Classes of Probability Distributions Based on the Concept of Reciprocal Coordinate Subtangent

1989 ◽  
Vol 38 (3-4) ◽  
pp. 169-180 ◽  
Author(s):  
S.P. Mukherjee ◽  
Dilip Roy
Keyword(s):  
Failure Rate ◽  
Probability Distributions ◽  
Life Distribution ◽  
Density Curve ◽  
Special Case

Tho usefulness of rociprocal coordinate subtancent as a measure on the density curve of a life distribution bas been pointed out through teveral results including some characterizations and classifications. Existing non­parametric classifications based on failure rate and avorage failure rate have been related to the nonparametric classifications proposed here. It bas also been sbown tbat a study through reciprocal coordinate subtangent results in more aeneral properties from which properties of extistiog classes can bo obtained as a special case.

scholarly journals On the Entropy of a One-Dimensional Gas with and without Mixing Using Sinai Billiard

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1188
Author(s):  
Alexander Sobol ◽  
Peter Güntert ◽  
Roland Riek
Keyword(s):  
Probability Distributions ◽  
Dimensional System ◽  
Strongly Correlated ◽  
Two Dimensional ◽  
One Dimensional ◽  
Point Particles ◽  
Sinai Billiard ◽  
Finite Space ◽  
The One ◽  
Special Case

A one-dimensional gas comprising N point particles undergoing elastic collisions within a finite space described by a Sinai billiard generating identical dynamical trajectories are calculated and analyzed with regard to strict extensivity of the entropy definitions of Boltzmann–Gibbs. Due to the collisions, trajectories of gas particles are strongly correlated and exhibit both chaotic and periodic properties. Probability distributions for the position of each particle in the one-dimensional gas can be obtained analytically, elucidating that the entropy in this special case is extensive at any given number N. Furthermore, the entropy obtained can be interpreted as a measure of the extent of interactions between molecules. The results obtained for the non-mixable one-dimensional system are generalized to mixable one- and two-dimensional systems, the latter by a simple example only providing similar findings.

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