Statistical fundamental theorem

This relationship is known as Liouville s equation. It is the fundamental theorem of statistical mechanics and is valid only for the phase space of generalized coordinates and momenta, whereas it is not, for example valid for the phase space of generalized coordinates and velocities. This is the basic reason that one chooses the Hamiltonian equations of motion in statistical mechanics. 

Now we show that there is a surprising relation between Fisher s fundamental theorem of natural selection and other theory developed by Fisher, the likelihood theory in statistics and Fisher information [21], As far as we know, the present chapter is the first publication in the literature pointing out the connections between these two problems formulated and studied by Fisher. 

Yet more important was the publication by Schottky and Wagner (1930) of their classical paper on the statistical thermodynamics of real crystals (41). This clarified the role of intrinsic lattice disorder as the equilibrium state of the stoichiometric crystal above 0° K. and led logically to the deduction that equilibrium between the crystal of an ordered mixed phase—i.e., a binary compound of ionic, covalent, or metallic type—and its components was statistical, not unique and determinate as is that of a molecular compound. As the consequence of a statistical thermodynamic theorem this proposition should be generally valid. The stoichiometrically ideal crystal has no special status, but the extent to which different substances may display a detectable variability of composition must depend on the energetics of each case—in particular, on the energetics of lattice disorder and of valence change. This point is taken up below, for it is fundamental to the problems that have to be considered. 

Over the past 15 years a number of important fundamental theorems in nonequilibrium statistical mechanics of many-particle systems have been proved. These proofs result in a number of important relations in nonequilibrium statistical mechanics. In this review we focus on three new relationships the dissipation function or Evans-Searles fluctuation relation (ES the Jarzynski equality... 

The subject of statistical mechanics is a branch of mechanics which has been found very useful in the discussion of the properties of complicated systems, such as a gas. In the following sections we shall give a brief discussion of the fundamental theorem of statistical quantum mechanics (Sec. 49a), its application to a simple system (Sec. 496), the Boltzmann distribution law (Sec. 49c), Fermi-Dirac and Bose-Einstein statistics (Sec. 49d), the rotational and vibrational energy of molecules (Sec. 49e), and the dielectric constant of a diatomic dipole gas (Sec. 49/). The discussion in these sections is mainly descriptive and elementary we have made no effort to carry through the difficult derivations or to enter into the refined arguments needed in a... 

This robustness arises from the central limit theorem, one of the fundamental theorems of statistics, which might be stated as follows ... 

Rather than characterizing the dynamics of a fluctuating state variable by the time-dependent correlation function (X (0)X (t)), one can also describe it by the spectral density Xy u ) ). The Wiener—Chinchin theorem, a fundamental theorem of statistical physics, states that these two functions represent a pair of Fourier transforms, i.e.. 

Assuming fj, < 1/2, this solution implies a monotonic approach to equilibrium with time. From a purely statistical point of view, this is certainly correct the difference in number between the two different balls decreases exponentially toward a state in which neither color is preferred. In this sense, the solution is consistent with the spirit of Boltzman s H-theorem, expressing as it does the idea of motion towards disorder. But the equation is also very clearly wrong. It is wrong because it is obviously inconsistent with the fundamental properties of the system it violates both the system s reversibility and periodicity. While we know that the system eventually returns to its initial state, for example, this possibility is precluded by equation 8.142. As we now show, the problem rests with equation 8.141, which must be given a statistical interpretation. 

There are a number of unsatisfactory features about this procedure which it is important to examine. The first is the uniqueness of the solution. From a fundamental viewpoint, we may believe that the Uniqueness Theorem in electromagnetism suggests that there is indeed only one possible perfect match between experiment and simulation. However, even if this is the case, we can never have sufficiently perfect data for this stringent condition to be valid. All data are intrinsically statistically noisy, have a non-zero background and a finite range of wavevector covered. In practice, there can be no traly unique solution and this immediately leads to the second problem, that of local minima. 

Frequentist methods are fundamentally predicated upon statistical inference based on the Central Limit Theorem. For example, suppose that one wishes to estimate the mean emission factor for a specific pollutant emitted from a specific source category under specific conditions. Because of the cost of collecting measurements, it is not practical to measure each and every such emission source, which would result in a census of the actual population distribution of emissions. With limited resources, one instead would prefer to randomly select a representative sample of such sources. Suppose 10 sources were selected. The mean emission rate is calculated based upon these 10 sources, and a probability distribution model could be fit to the random sample of data. If this process is repeated many times, with a different set of 10 random samples each time, the results will vary. The variation in results for estimates of a given statistic, such as the mean, based upon random sampling is quantified using a sampling distribution. From sampling distributions, confidence intervals are obtained. Thus, the commonly used 95% confidence interval for the mean is a frequentist inference... 

The aim of this chapter is to show how the concepts of FDT violation and effective temperature can be illustrated in the framework of the above quoted system, as done experimentally in Ref. 12 and theoretically in Refs. 15-19. We do not discuss here the vast general domain of aging effects in glassy systems, which are reviewed in Refs. 2-4. Since the present contribution should be understood by beginners in the field, some relevant fundamental topics of equilibrium statistical physics—namely, on the one hand, the statistical description of a system coupled to an environment and, on the other hand, the fluctuation-dissipation theorem (in a time domain formulation)—are first recalled. Then, questions specifically related to out-of-equilibrium dynamics, such as the description of aging effects by means of an effective temperature, are taken up in the framework of the above-quoted model system. 

This chapter discusses several statistical mechanical theories that are strongly positioned in the historical sweep of the theory of liquids. They are chosen for inclusion here on the basis of their potential for utility in analyzing simulation calculations, and their directness in conneeting to the other fundamental topic discussed in this book, the potential distribution theorem. Therefore tentacles can be understood as tentacles of the potential distribution theorem. From the perspective of the preface discussion, the theories presented here might be useful for discovery of models such as those discussed in Chapter 4. These theories are a significant subset of those referred to in Chapter 1 as ... both difficult and strongly established. .. (Friedman and Dale, 1977), but the present chapter does not exhaust the interesting prior academic development of statistical mechanical theories of solutions. Sections 6.2 and 6.3 discuss alternative views of chemical potentials, namely those of density functional theory and fluctuation theory. 

A fundamental hypothesis of statistical mechanics is the ergodic theorem. Basically it says the system evolves so quickly in the phase space that it visits all of the possible phase points during the time considered. If the system is eigodic, the ensemble average is equivalent to the time average over the trajectory for the time period. The ergodidty of a system depends on the search procedure, force... 

In the Hamiltonian formulation the Liouville equation can be seen as a continuity or advection equation for the probability distribution function. This theorem is fundamental to statistical mechanics and requires further attention. 

We also note the equality of time and ensemble averages which is a fundamental tenet of statistical mechanics (ergodic theorem)... 

Thus, one is interested in a rather special kind of statistics, viz. the statistics of a dense population of interacting levels. This is the fundamental distinction between chaos and complexity there may arise situations in which levels do not necessarily all interact (they might have different quantum numbers) but are simply present in large numbers, so that their analysis is not possible in practice but could be performed in principle. These are called unresolved transition arrays (UTAs). One can develop [526] a theory of UTAs which yields general theorems about them as a whole. Such theories are a statistical approach to the interpretation of spectra, but are not related to the problem of quantum chaology. 

There are many different ways to treat mathematically uncertainly, but the most common approach used is the probability analysis. It consists in assuming that each uncertain parameter is treated as a random variable characterised by standard probability distribution. This means that structural problems must be solved by knowing the multi-dimensional Joint Probability Density Function of all involved parameters. Nevertheless, this approach may offer serious analytical and numerical difficulties. It must also be noticed that it presents some conceptual limitations the complete uncertainty parameters stochastic characterization presents a fundamental limitation related to the difficulty/impossibility of a complete statistical analysis. The approach cannot be considered economical or practical in many real situations, characterized by the absence of sufficient statistical data. In such cases, a commonly used simplification is assuming that all variables have independent normal or lognormal probability distributions, as an application of the limit central theorem which anyway does not overcome the previous problem. On the other hand the approach is quite usual in real situations where it is only possible to estimate the mean and variance of each uncertainty parameter it being not possible to have more information about their real probabilistic distribution. The case is treated assuming that all uncertainty parameters, collected in the vector d, are characterised by a nominal mean value iJ-dj and a correlation =. In this specific... 

In the Hamiltonian formulation, the Liouville equation can be seen as a continuity or advection equation for the probability distribution function. This theorem is fundamental to statistical mechanics and requires further attention. Considering an ensemble of initial conditions each representing a possible state of the system, we express the probability of a given ensemble or density distribution of system points... 

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