scholarly journals Symmetric matrix ensemble and integrable hydrodynamic chains

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Costanza Benassi ◽  
Marta Dell’Atti ◽  
Antonio Moro
Keyword(s):  
Partition Function ◽  
Thermodynamic Limit ◽  
Arbitrary Number ◽  
Symmetric Matrix ◽  
Hydrodynamic Type ◽  
Number Of Components ◽  
Matrix Ensemble ◽  
Particular Solution ◽  
Hydrodynamic Reductions ◽  
Hydrodynamic Chain

AbstractThe partition function of the Symmetric Matrix Ensemble is identified with the $$\tau $$ τ -function of a particular solution of the Pfaff Lattice. We show that, in the case of even power interactions, in the thermodynamic limit, the $$\tau $$ τ -function corresponds to the solution of an integrable chain of hydrodynamic type. We prove that the hydrodynamic chain so obtained is diagonalisable and admits hydrodynamic reductions in Riemann invariants in an arbitrary number of components.

scholarly journals THE CRITICAL POINT OF UNORIENTED RANDOM SURFACES WITH A NONEVEN POTENTIAL

1993 ◽  
Vol 08 (06) ◽  
pp. 1139-1152
Author(s):  
M.A. MARTÍN-DELGADO
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Critical Point ◽  
Discrete Model ◽  
Recursion Relation ◽  
Scaling Limit ◽  
Random Surfaces ◽  
Matrix Ensemble ◽  
Quartic Interaction ◽  
Double Scaling Limit

The discrete model of the real symmetric one-matrix ensemble is analyzed with a cubic interaction. The partition function is found to satisfy a recursion relation that solves the model. The double scaling-limit of the recursion relation leads to a Miura transformation relating the contributions to the free energy coming from oriented and unoriented random surfaces. This transformation is the same kind as found with a quartic interaction.

scholarly journals TWO-DIMENSIONAL YANG-MILLS THEORIES ARE STRING THEORIES

1993 ◽  
Vol 08 (23) ◽  
pp. 2223-2235 ◽  
Author(s):  
STEPHEN G. NACULICH ◽  
HAROLD A. RIGGS ◽  
HOWARD J. SCHNITZER
Keyword(s):  
String Theory ◽  
Partition Function ◽  
Arbitrary Number ◽  
Closed String ◽  
Two Dimensional ◽  
Branched Covers ◽  
Yang Mills ◽  
String Worldsheet ◽  
String Theories

We show that two-dimensional SO (N) and Sp (N) Yang-Mills theories without fermions can be interpreted as closed string theories. The terms in the 1/N expansion of the partition function on an orientable or nonorientable manifold ℳ can be associated with maps from a string worldsheet onto ℳ. These maps are unbranched and branched covers of ℳ with an arbitrary number of infinitesimal worldsheet cross-caps mapped to points in ℳ. These string theories differ from SU (N) Yang-Mills string theory in that they involve odd powers of 1/N and require both orientable and nonorientable worldsheets.

PARTITION FUNCTION OF THE GAS-LIQUID COEXISTENT SYSTEM

2007 ◽  
Vol 21 (21) ◽  
pp. 3755-3764
Author(s):  
SUHONG ZHANG ◽  
ZIJING LI ◽  
YUANXING GUI ◽  
WEI WANG
Keyword(s):  
Partition Function ◽  
Thermodynamic Limit ◽  
Particle System ◽  
Phase Region ◽  
Horizontal Line ◽  
Fluid System ◽  
Two Phase ◽  
The One ◽  
Multi Phase ◽  
Finite Particle

In this article the equilibrious gas-liquid coexistent system is studied, and a new expression of partition function (PF) corresponding to the two-phase region is derived. Based on this expression, the horizontal line in the isotherm of pressure versus volume is obtained naturally for a finite particle system (i.e., without the necessity of taking the thermodynamic limit). Extending this PF, we can gain a unitive form of the one-component fluid in any system (i.e., one-phase or multi-phase). Then the whole isotherm will have reasonable statistical foundation. The VDW fluid system is discussed as a concrete example.

scholarly journals Modeling Spacetime as a Branched Covering Space over 2-Knots

2019 ◽  
Author(s):  
Christopher Duston
Keyword(s):  
Gravitational Field ◽  
Partition Function ◽  
Thermodynamic Limit ◽  
Dimensional Reduction ◽  
Degrees Of Freedom ◽  
Covering Space ◽  
Topological Information ◽  
Spacetime Geometry ◽  
Branched Covering

In this paper we review a proposal to represent the geometric degrees of freedom of the gravitational field as a branched covering space, and introduce a new application of this in which the branch loci are 1- or 2-knots. This allows one to construct arbitrary smooth, closed 3- and 4-manifolds with enough geometric and topological information to write down a partition function and calculate statistical quantities in the thermodynamic limit. Further, we find clear evidence for a dimensional reduction of the spacetime geometry from four to two. As an example, we choose a family of smooth 4-manifolds presented in this way, and calculate the entropy of the system.

scholarly journals Exact Solution of a One-dimensional Spin System

1970 ◽  
Vol 23 (5) ◽  
pp. 927 ◽  
Author(s):  
RW Gibberd
Keyword(s):  
Phase Transition ◽  
Exact Solution ◽  
Free Energy ◽  
Partition Function ◽  
Thermodynamic Limit ◽  
Spin System ◽  
Spin Systems ◽  
One Dimensional ◽  
Gibb’S Free Energy ◽  
Theory Of Superconductivity

The partition function and the Gibb's free energy are calculated exactly in the thermodynamic limit, using techniques which are well known in the theory of superconductivity. This calculation illustrates explicitly the similarity between the phase transition in superconductivity and the molecular field transitions in spin systems.

Contributions to the statistical theory of adsorption. II

1938 ◽  
Vol 34 (4) ◽  
pp. 587-598 ◽  
Author(s):  
G. P. Dube
Keyword(s):  
Statistical Theory ◽  
Partition Function ◽  
Adsorption Isotherm ◽  
Critical Phenomena ◽  
General Problem ◽  
Arbitrary Number ◽  
Solid Surfaces ◽  
Attractive Force ◽  
Adsorbed Molecules ◽  
Number Of Layers

The adsorption of gaseous molecules in a monolayer on solid surfaces has been studied theoretically by several workers, namely Fowler, Peierls and Wang. They have shown that the adsorption isotherm exhibits critical phenomena if there is assumed to be an attractive force between the neighbouring adsorbed molecules, and this has provided an explanation of the critical condensation phenomena observed in deposition experiments, for example those of Cockcroft on the deposition of cadmium on copper. The actual critical phenomena observed consist in the deposition of many layers which are formed if the first layer has fairly started. It is of interest, therefore, to consider if there are other ways in which critical phenomena can arise than through interactions in the first layer. The general problem of adsorption of an arbitrary number of layers is too complicated to study because of the mathematical difficulty in constructing the partition function. We therefore confine ourselves to the consideration of adsorption in two layers only.

GENERALIZED PARTITION FUNCTION FOR BOSE STATISTICS AND THERMODYNAMIC FUNCTIONS

1999 ◽  
Vol 13 (29n30) ◽  
pp. 1055-1062 ◽  
Author(s):  
HONG-YI FAN ◽  
HUI ZOU
Keyword(s):  
Partition Function ◽  
Coherent State ◽  
Thermodynamic Functions ◽  
Nonlinear Medium ◽  
Symmetric Matrix ◽  
State Representation ◽  
Bose Distribution ◽  
Bose Statistics ◽  
Photon Gas ◽  
Generalized Partition

Based on the density operator's coherent state representation [Phys. Lett.A252, 281 (1999)], we extend the well-known Bose distribution's partition function Ξ [Formula: see text] for ideal photon gas [Formula: see text] to that of an assembly of photon gas interacting with nonlinear medium described by the Hamiltonian [Formula: see text], where A=(a1,a2,…,an), Ω is a symmetric matrix, and we report that the corresponding partition function is [Formula: see text] As a result, the generalized Bose distribution and thermodynamic functions are derived.

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