Darwin-Fowler Method Research Articles

On the Partition Temperature of Massless Particles in High-Energy Collisions

Although partition temperature derived using the Darwin–Fowler method is exact for simple scenarios, the derivation for complex systems might reside in specific approximations whose viability is not ensured if the thermodynamic limit is not attained. This work elaborates on a related problem relevant to relativistic high-energy collisions. On the one hand, it is simple enough that closed-form expressions can be obtained precisely for the one-particle distribution function. On the other hand, the resulting expression is not an exponential form, and therefore, it is not straightforward that the notion of partition function could be implied. Specifically, we derive the one-particle distribution function for massless particles where the phase-space integration is performed exactly for the underlying canonical ensemble consisting of a given number of particles. We discuss the viability of the partition temperature in this case. Possible implications of the obtained results regarding the observed Tsallis distribution in transverse momentum spectra in high-energy collisions are also addressed.

The Canonical Ensemble of Open Self-Avoiding Strings

Statistical models of a single open string avoiding self-intersections in the d-dimensional Euclidean space ℝd, 2 ≤ d < 4, and the ensemble of strings are considered. The presentation of these models is based on the Darwin–Fowler method, used in statistical mechanics to derive the canonical ensemble. The configuration of the string in space ℝd is described by its contour length L and the spatial distance R between its ends. We establish an integral equation for a transformed probability density W(R, L) of the distance R similar to the known Dyson equation, which is invariant under the continuous group of renormalization transformations. This allows us using the renormalization group method to investigate the asymptotic behavior of this density in the case where R → ∞ and L → ∞. For the model of an ensemble of M open strings with the mean string contour length over the ensemble given by $$ \overline{L} $$ , we obtain the most probable distribution of strings over their lengths in the limit as M → ∞. Averaging the probability density W(R, L) over the canonical ensemble eventually gives the sought density 〈W(R, L)〉.

A d-dimensional model of the canonical ensemble of open strings

We propose a d-dimensional model of the canonical ensemble of open self-avoiding strings. We consider the model of a solitary open string in the d-dimensional Euclidean space ℝ d, 2 ≤ d < 4, where the string configuration is described by the arc length L and the distance R between string ends. The distribution of the spatial size of the string is determined only by its internal physical state and interaction with the ambient medium. We establish an equation for a transformed probability density W(R,L) of the distance R similar to the known Dyson equation, which is invariant under the continuous group of renormalization transformations; this allows using the renormalization group method to investigate the asymptotic behavior of this density in the case where R→∞ and L→∞. We consider the model of an ensemble of M open strings with the mean string length over the ensemble given by $\bar L$ , and we use the Darwin-Fowler method to obtain the most probable distribution of strings over their lengths in the limit as M →∞. Averaging the probability density W(R,L) over the canonical ensemble eventually gives the sought density 〈W(R, $\bar L$ )〉.

Clustering in coagulation-fragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and limiting laws

We develop a unified approach to the problem of clustering in the three different fields of applications indicated in the title of the paper, in the case when the parametric function of the models is regularly varying with positive exponent. The approach is based on Khintchine’s probabilistic method that grew out of the Darwin-Fowler method in statistical physics. Our main result is the derivation of asymptotic formulae for the distribution of the largest and the smallest clusters (= components), as the total size of a structure (= number of particles) goes to infinity. We discover that n 1 l + 1 n^{\frac {1}{l+1}} is the threshold for the limiting distribution of the largest cluster. As a by-product of our study, we prove the independence of the numbers of groups of fixed sizes, as n → ∞ . n\to \infty . This is in accordance with the general principle of asymptotic independence of sites in mean-field models. The latter principle is commonly accepted in statistical physics, but not rigorously proved.

Electronic noise due to multiple trap levels in homogeneous solids and in space-charge layers

A near-exact expression is derived for carrier-density fluctuations associated with multiple trap levels, interacting with one band only (say, the conduction band), using the many-variate Master Equation. The canonical constraint is handled with the Darwin–Fowler method. For a homogeneous solid with a discrete number of traps the spectrum consists of a sum of Lorentzians, as expected. For the case of a continuously distributed energy range, the results do not support a Bernamont–Surdin–McWhorter 1/f-like envelope spectrum; on the contrary, no more than two decades of 1/f noise can be expected. The situation is different, however, when band-bending occurs. Employing a Green’s function approach, developed by us previously, we give a formal complete result, which can be evaluated when Ec(r) is known.

Complexation to macromolecules with a large number of sites

This paper presents an approach based on the saddle-point approximation to study the equilibrium interactions between small molecules and macromolecules with a large number of sites. For this case, the application of the Darwin–Fowler method results in very simple expressions for the stoichiometric equilibrium constants and their corresponding free energies in terms of integrals of the binding curve plus a correction term which depends on the first derivatives of the binding curve in the points corresponding to an integer value of the mean occupation number. These expressions are simplified when the number of sites tends to infinity, providing an interpretation of the binding curve in terms of the stoichiometric stability constants. The formalism presented is applied to some simple complexation models, obtaining good values for the free energies involved. When heterogeneous complexation is assumed, simple expressions are obtained to relate the macroscopic description of the binding, given by the stoichiomeric constants, with the microscopic description in terms of the intrinsic stability constants or the affinity spectrum.

The nuclear level-density parameter and temperature-dependent effects in finite nuclei

Abstract The influence of temperature-dependent effects on nuclear level densities is investigated. The explicit treatment of thermal degrees of freedom, within the Darwin-Fowler method, shows that the level-density parameter strongly differs from a constant value at low temperatures. The formalism is presented both for normal and superfluid systems.

Single-particle momentum distribution at high energies and concept of partition temperature.

A concept of partition temperature is introduced in high-energy collisions. It is a natural mathematical consequence of the Darwin-Fowler method, and neither requires nor implies thermal equilibrium. A collision at a given incoming energy is described as an incoherent superposition of collisions with different partition temperatures. Angular distributions are then presented for ..sqrt..s = 540 GeV collisions.

Nuclear level densities

Nuclear level densities are obtained by first calculating the density for non-interacting particles using the Darwin–Fowler method and then folding in approximately the effects of the residual two-body interaction. For the former, the equations resulting from the method of steepest descent are solved numerically with a realistic set of single particle energies and the latter is then incorporated using statistical spectroscopy considerations. Good agreement with experimental data is found.

An information theoretical approach to inversion problems

An inversion procedure which provides the most conservative inference for an unknown function in terms of partial data is discussed on the basis of information theoretic considerations. The method is based on the procedure of maximal entropy, but is not limited to the estimation of unknown probabilities. Rather, inductive inferences can be drawn regarding the values of general (if necessary, dimension-bearing) variables. The solution of an inversion problem using data linear in the unknown function is discussed in detail and explicit results are obtained. For every class of problems with common symmetry properties, the general algorithm can be reduced to a more direct procedure. When the data consist of average values for an unknown distribution, the general approach is in the spirit of the Darwin-Fowler method, while the reduced route is the procedure of maximal entropy ('method of most probable distribution') as usually employed in statistical mechanics. Other classes of problems discussed include the representation of an unknown function in a complete orthonormal basis using as input a partial set of expansion coefficients, and the inference of line shapes and power spectra.

A statistical derivation of the density of final states for the exciton model

A new method for the derivation of the density of final states in the nuclear equilibration process, based on the Darwin-Fowler method of statistical physics, is suggested. This method is generally valid for an arbitrary energy level scheme and includes the Pauli principle. When it is applied to the equidistant-spacing energy level scheme, the results are very close to those derived by Obložinský et al.

Geometric programming and the Darwin-Fowler method in statistical mechanics

Geometric programming and the Darwin-Fowler method in statistical mechanics

Inversion of the Partition Function: The First-Order Steepest-Descent Method

This paper explains and advocates the use of a Laplace-transform steepest-descent method for the computation of energy-level densities g(E) by inversion of the partition function. The earlier Lin–Eyring approach, based on the Darwin–Fowler method, is shown to be unnecessarily obscure and in part redundant By use of a much-simpler formulation due to Kubo, it is shown that the first-order steepest-descent approximation to the inversion integral follows straightforwardly from a knowledge of the relevant partition function, its energy, and specific-heat relations. In this way several explicit equations for g(E) and its integral W(E) are derived, including two for anharmonic vibration and internal vibration–rotation. The relationship of the S–D method to Tolman's classical formula is discussed and some thermodynamic implications pointed out. Numerical results agree closely with those obtained by the direct counting of energy levels and, with certain exceptions, the indirect computations of other authors.

On the connection between quantum-mechanical green’s functions and the darwin-fowler method in statistical mechanics

It is proved in a general ay how the generating functions which are used in Statistical Mechanics is connection with the Darwin-Fowler method can be obtained in a simple manner as spurs of Green Kernels.

Densité de niveaux d'un système de nucléons en interaction dans une couche

Using perturbation theory, the grand partition function is calculated for the completely degenerate system of interacting nucleons inside a shell. The level density is then derived by the Darwin-Fowler method. The purpose is to compare these results to exact levels known for a p-shell. The method gives fairly good results, even for a small number of particles, provided that the self-consistent field is taken into account, and the calculation is carried out in a coherent way; there is no advantage in performing the saddle point method after integration of the level density.

A general kinetic theory of liquids. IV. Quantum mechanics of fluids

In this paper the classical theory of liquids developed in the first three parts of this series is translated into the quantum formalism. After the fundamental equations have been reformulated, it is shown that in the classical limit ℏ = 0, they go over into the corresponding classical equations. A quantized proof of the Boltzmann distribution law is given which is simpler and more direct than that of the Darwin-Fowler method. Then the equation of state is derived in a form which exhibits clearly the deviations from the classical law at very low temperatures. An approximate method of solution of the fundamental equations is developed in a form suitable for practical application. Finally, the quantum equations of motion and energy transport are obtained, and it is shown that they are formally identical with the classical ‘hydrodynamical equations’. This enables a discussion of the viscosity and thermal conductivity of quantum liquids to be given which exposes clearly the alternative explanations of the abnormal properties of liquid He II.