Meinardus Theorem Research Articles

A Meinardus Theorem with Multiple Singularities

Meinardus proved a general theorem about the asymptotics of the number of weighted partitions, when the Dirichlet generating function for weights has a single pole on the positive real axis. Continuing \cite{GSE}, we derive asymptotics for the numbers of three basic types of decomposable combinatorial structures (or, equivalently, ideal gas models in statistical mechanics) of size $n$, when their Dirichlet generating functions have multiple simple poles on the positive real axis. Examples to which our theorem applies include ones related to vector partitions and quantum field theory. Our asymptotic formula for the number of weighted partitions disproves the belief accepted in the physics literature that the main term in the asymptotics is determined by the rightmost pole.

A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions

A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions

A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions

An asymptotic formula for the number of states of Boson gas whose Hamiltonian is given by a positive elliptic pseudo-differential operator of order one on a compact manifold is given under a integrality assumption on the spectrum of the Hamiltonian. This is regarded as an analogue of the Meinardus theorem on asymptotics of the number of partitions of a positive integer.

Thermodynamic properties of the 2N-piece relativistic string

The thermodynamic free energy F(β) is calculated for a gas consisting of the transverse oscillations of a piecewise uniform bosonic string. The string consists of 2N parts of equal length, of alternating type I and type II material, and is relativistic in the sense that the velocity of sound everywhere equals the velocity of light. The present paper is a continuation of two earlier papers, one dealing with the Casimir energy of a 2N-piece string [I. Brevik and R. Sollie, J. Math. Phys. 38, 2774 (1997)], and another dealing with the thermodynamic properties of a string divided into two (unequal) parts [I. Brevik, A. A. Bytsenko, and H. B. Nielsen, Class. Quantum Grav. 15, 3383 (1998)]. Making use of the Meinardus theorem, we calculate the asymptotics of the level state density, and show that the critical temperatures in the individual parts are equal, for arbitrary space–time dimension D. If D=26, we find β=(2/N)2π/TII, TII being the tension in part II. Thermodynamic interactions of parts related to high genus g is also considered.

Infinite products, partition functions, and the Meinardus theorem

The Meinardus theorem provides an asymptotic value for the infinite-product generating functions of many different partition functions, and thereby asymptotic values for these partition functions themselves. Under assumptions somewhat stronger than those of Meinardus a more specific version of the theorem is derived, then this is generalized from the spectrum of integers considered by Meinardus to infinite products constructed from an arbitrary spectrum. Included are infinite products that generate partition functions able to count states in the quantum theory of extended objects. There emerges a quite general and explicit method (of which only limited details are given here) for deriving the asymptotic growth of the state density function of multidimensional quantum objects. One finds a universal geometry-independent leading behavior, with nonleading geometry-dependent corrections that the method is capable of providing explicitly.

Critical temperature in the thermodynamics of extended objects

Extending previous work on the thermodynamics of quantum strings a low-temperature power series expansion is derived with the help of the Meinardus theorem for the free energy of a quantum p-brane having toroidal compactification. In the string case p = 1 this series has a finite radius of convergence corresponding to the well-known Hagedorn temperature. For p>1 the radius of convergence drops to zero. However, for odd p it is possible to resum unambiguously the divergent power series in temperature T to a form in which T is unrestricted. This suggests that odd p-branes at least there may be no limiting temperature, and moreover there may exist a fundamental difference between even and odd p-branes.

Meinardus theorem and the asymptotic form of the quantum p-brane state density

Within the semiclassical approximation for the p-branes (discrete mass spectrum) the asymptotic form of the density of states is obtained. The universal feature of the density, which grows slower than for black holes but faster than the one for strings, is found by using rigorous results due to Meinardus. In particular the explicit forms of the prefactors of the p-branes total level degeneracy are calculated. A brief physical discussion about the thermodynamic properties of an extended object is also presented.