Unnormalized probability: A different view of statistical mechanics
نویسنده
چکیده
All teachers and students of physics have absorbed the doctrine that probability must be normalized. Nevertheless, there are problems for which the normalization factor only gets in the way. An important example of this counter-intuitive assertion is provided by the derivation of the thermodynamic entropy from the principles of statistical mechanics. Unnormalized probabilities provide a surprisingly effective teaching tool that can make it easier to explain to students the essential concept of entropy. The elimination of the normalization factor offers simpler equations for thermodynamic equilibrium in statistical mechanics, which then lead naturally to a new and simpler definition of the entropy in thermodynamics. Notably, this definition does not change the formal expression of the entropy based on composite systems that I have previously offered. My previous definition of entropy has been criticized by Dieks, based on what appears to be a misinterpretation. I believe that the new definition presented here has the advantage of greatly reducing the possibility of such a misunderstanding—either by students or by experts.
منابع مشابه
An Alternative Interpretation of Probability Measures in Statistical Mechanics
I offer an alternative interpretation of classical statistical mechanics and the role of probability in the theory. In my view the stochasticity of statistical mechanics is associated directly with the observables rather than microstates. This view requires taking seriously the idea that the physical state of a statistical mechanical system is a probability measure, thereby avoiding the unneces...
متن کاملBoundary conditions for probability density function transport equations in fluid mechanics.
The behavior of the probability density function (PDF) transport equation at the limits of the probability space is studied from the point of view of fluid mechanics. Different boundary conditions are considered depending on the nature of the variable considered (velocity, scalar, and position). A study of the implications of entrance and exit conditions is performed, showing that a new term sh...
متن کاملLimits of Spectral Clustering
An important aspect of clustering algorithms is whether the partitions constructed on finite samples converge to a useful clustering of the whole data space as the sample size increases. This paper investigates this question for normalized and unnormalized versions of the popular spectral clustering algorithm. Surprisingly, the convergence of unnormalized spectral clustering is more difficult t...
متن کاملNoise-Contrastive Estimation of Unnormalized Statistical Models, with Applications to Natural Image Statistics
We consider the task of estimating, from observed data, a probabilistic model that is parameterized by a finite number of parameters. In particular, we are considering the situation where the model probability density function is unnormalized. That is, the model is only specified up to the partition function. The partition function normalizes a model so that it integrates to one for any choice ...
متن کاملNormalized Scale-Space Derivatives: A Statistical Analysis
This chapter presents a statistical analysis of multiscale derivative measurements. Noisy images and multiscale derivative measurements made of noisy images are analyzed; the means and variances of the measured noisy derivatives are calculated in terms of the parameters of the probability distribution function of the initial noise function and the scale or sampling aperture. Normalized and unno...
متن کامل