M. Howard Lee

Derivation of the generalized Langevin equation by a method of recurrence relations

The generalized Langevin equation was first derived by Mori using the Gram‐Schmidt orthogonalization process. This equation can also be derived by a method of recurrence relations. For a physical space commonly used in statistical mechanics, the recurrence relations are simple and they lead directly to the Langevin equation. The Langevin equation is shown to be composed of one homogeneous and one inhomogeneous equation.

Polylogarithmic analysis of chemical potential and fluctuations in a D‐dimensional free Fermi gas at low temperatures

The chemical potential and fluctuations in number of particles in a D‐dimensional free Fermi gas at low temperatures are obtained by means of polylogarithms. This idea is extended to show that the density of any ideal gas, whether Fermi, Bose, or classical, can be expressed in polylogarithms. The densities of different statistics correspond to different domains of polylogarithms in such a way that there emerges a unifying picture. The density of the classical ideal gas represents a fixed point of polylogarithms. Inequalities for polylogarithms are used to provide a precise bound on errors in the fermion chemical potential at low temperatures.

Method of Recurrence Relations and Applications to Many-Body Systems

The method of recurrence relations allows one to calculate time dependent correlation functions from first principles. It originates from the Kubo scalar product which realizes abstract Hilbert space of dynamical variables. For this realized space, we show that there exists a unique orthogonalization process by recurrence relation. The method of recurrence relations exploits geometric properties of the structure of the realized Hilbert space such as the dimensionality and shape. These geometric properties depend on models as well as static constraints, e.g., temperature, wave vector, size. This method has been applied to several physical models including the homogeneous dense electron gas, the spin-1/2 XY and transverse Ising models, the spin van der Waals model and others. For these models, we have obtained the relaxation function, response function and memory function.

Chemical potential of a D‐dimensional free Fermi gas at finite temperatures

The chemical potential of a D‐dimensional free Fermi gas at low temperatures has been obtained using some ideas due to Barker and Blankenbechler [J. Math. Phys. 9 7, 302 (1986); Am. J. Phys. 2 5, 279 (1957)]. There is an interesting even–odd‐dimensional effect in the behavior of the chemical potential, also observed in other properties of a free Fermi gas, such as the susceptibility. For D=2, it is possible to give a closed form expression, thus valid both in the high‐ and low‐temperature regions, probably not possible in any other even or odd dimensions.

Analytical study of the superstable 3-cycle in the logistic map

In the logistic map, the simplest of stable odd-numbered cycles is a 3-cycle. It comes into existence after the stable 2k cycles cease to exist. The 3-cycle is thus important as a route to chaos. In 1977 Guckenheimer et al. [J. Math. Biol. 4, 101 (1977)] first estimated the value of the control parameter a at which it can be superstable. It implies that a stable 3-cycle can be formed and deformed at some values of a straddling that of the superstable 3-cycle. In this work we present an exact value of a for the superstable 3-cycle by reducing a polynomial of degree 7 to that of 3. The relevance of this work to Sharkovskii’s theorem is also discussed.

Wave vector dependent susceptibility of a free electron gas in D dimensions and the singularity at 2kF

The static susceptibility of a free electron gas in D dimensions at T=0 is obtained by techniques of dimensional regularization. Our solutions for the susceptibility χ(k,D) are given in terms of the hypergeometric function. For any integer dimensions analytic expressions are possible. The high‐ and low‐k series solutions are shown to be related by an analytic continuation if D is an odd integer, but not related if D is an even integer. The singularity at 2kF is a branch point, whereupon the series solutions are absolutely convergent, yielding χ(k=2kF,D)=(D−1)−1. The relationship of χkD has the appearance of a PVT diagram.

Long‐range order in the spin van der Waals model

Long‐range order in the spin van der Waals model is considered when the number of spins is finite and also when infinite. We show explicitly that a finite system cannot support long‐range order. An infinite system at high temperatures is found to be dominated by the entropy of degenerate states of the system and, as a result, the system behaves essentially like an ideal system. In an infinite system at low temperatures, long‐range order exists, fully reflecting the spin symmetry of the Hamiltonian. For the XY‐like regime (J≳Jz ), mx is finite but mz vanishes, where mx and mz are reduced order parameters for the transverse and longitudinal directions, respectively. For the Ising‐like regime (J<Jz), mx vanishes but mz is finite. The isotropic interaction (Jz = J) behaves as a singularity and it must be considered separately. A physical interpretation of the behavior of long‐range order is offered using the geometry of spin space.

High‐Temperature Expansion of the Spin‐½ XY Model

The high‐temperature expansion of the spin‐½ XY model is shown to be particularly simple when the Hamiltonian is written in the second‐quantization form. By aid of a few simple rules, the partition function and susceptibility are easily evaluated to high orders.

Carnot cycle for photon gas

The Carnot cycle for a photon gas provides a useful means to illustrate the thermodynamic laws. It is particularly useful in showing the path dependence of thermodynamic functions. Thermodynamic relationships to a neutrino gas are also drawn.

Ergodicity in simple and not so simple systems and Kubo's condition

The ergodic hypothesis is approached from the side of time averaging to ascertain regions of its validity. A necessary and sufficient condition is formulated in terms of certain infinite products and tested in models of nn coupled harmonic oscillators. The validity of the hypothesis requires energy delocalizability, indicated by the existence of coherent translation modes. Kubos condition is re-examined to see the origin of its limitations.