Counting without sampling: Asymptotics of the log-partition function for certain statistical physics models

The evaluation of partition functions is a central problem in statistical physics. For lattice systems and other discrete models the partition function may be expressed as the contraction of a tensor network. Unfortunately computing such contractions is difficult, and many methods to make this tractable require periodic or otherwise structured networks. Here I present a new algorithm for contracting unstructured tensor networks. This method makes no assumptions about the structure of the network and performs well in both structured and unstructured cases so long as the correlation structure is local.

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Effect of Savings on a Gas-Like Model Economy with Credit and Debt

Entropy ◽

10.3390/e23020196 ◽

2021 ◽

Vol 23 (2) ◽

pp. 196

Author(s):

Guillermo Chacón-Acosta ◽

Vanessa Ángeles-Sánchez

Keyword(s):

Economic Growth ◽

Partition Function ◽

Statistical Physics ◽

Wealth Distribution ◽

Economic Inequality ◽

Economic Literature ◽

Payment Methods ◽

Income And Wealth Distribution ◽

Ensemble Formalism

In kinetic exchange models, agents make transactions based on well-established microscopic rules that give rise to macroscopic variables in analogy to statistical physics. These models have been applied to study processes such as income and wealth distribution, economic inequality sources, economic growth, etc., recovering well-known concepts in the economic literature. In this work, we apply ensemble formalism to a geometric agents model to study the effect of saving propensity in a system with money, credit, and debt. We calculate the partition function to obtain the total money of the system, with which we give an interpretation of the economic temperature in terms of the different payment methods available to the agents. We observe an interplay between the fraction of money that agents can save and their maximum debt. The system’s entropy increases as a function of the saved proportion, and increases even more when there is debt.

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Bounding the Partition Function of Spin-Systems

The Electronic Journal of Combinatorics ◽

10.37236/1098 ◽

2006 ◽

Vol 13 (1) ◽

Cited By ~ 4

Author(s):

David J. Galvin

Keyword(s):

Ising Model ◽

Partition Function ◽

Probability Distribution ◽

Statistical Physics ◽

Hard Core ◽

Spin Systems ◽

Normalizing Constant ◽

Graph Homomorphisms ◽

Hard Core Model ◽

Regular Bipartite Graph

With a graph $G=(V,E)$ we associate a collection of non-negative real weights $\bigcup_{v\in V}\{\lambda_{i,v}:1\leq i \leq m\} \cup \bigcup_{uv \in E} \{\lambda_{ij,uv}:1\leq i \leq j \leq m\}.$ We consider the probability distribution on $\{f:V\rightarrow\{1,\ldots,m\}\}$ in which each $f$ occurs with probability proportional to $\prod_{v \in V}\lambda_{f(v),v}\prod_{uv \in E}\lambda_{f(u)f(v),uv}$. Many well-known statistical physics models, including the Ising model with an external field and the hard-core model with non-uniform activities, can be framed as such a distribution. We obtain an upper bound, independent of $G$, for the partition function (the normalizing constant which turns the assignment of weights on $\{f:V\rightarrow\{1,\ldots,m\}\}$ into a probability distribution) in the case when $G$ is a regular bipartite graph. This generalizes a bound obtained by Galvin and Tetali who considered the simpler weight collection $\{\lambda_i:1 \leq i \leq m\} \cup \{\lambda_{ij}:1 \leq i \leq j \leq m\}$ with each $\lambda_{ij}$ either $0$ or $1$ and with each $f$ chosen with probability proportional to $\prod_{v \in V}\lambda_{f(v)}\prod_{uv \in E}\lambda_{f(u)f(v)}$. Our main tools are a generalization to list homomorphisms of a result of Galvin and Tetali on graph homomorphisms and a straightforward second-moment computation.

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The spectrum of an asymmetric annihilation process

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2884 ◽

2010 ◽

Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽

Author(s):

Arvind Ayyer ◽

Volker Strehl

Keyword(s):

Partition Function ◽

Recent Work ◽

Transition Matrix ◽

Statistical Physics ◽

Original Model ◽

Annihilation Process ◽

Generalized Model ◽

Nonequilibrium Statistical Physics ◽

International Audience ◽

The One

International audience In recent work on nonequilibrium statistical physics, a certain Markovian exclusion model called an asymmetric annihilation process was studied by Ayyer and Mallick. In it they gave a precise conjecture for the eigenvalues (along with the multiplicities) of the transition matrix. They further conjectured that to each eigenvalue, there corresponds only one eigenvector. We prove the first of these conjectures by generalizing the original Markov matrix by introducing extra parameters, explicitly calculating its eigenvalues, and showing that the new matrix reduces to the original one by a suitable specialization. In addition, we outline a derivation of the partition function in the generalized model, which also reduces to the one obtained by Ayyer and Mallick in the original model. Dans un travail récent sur la physique statistique hors équilibre, un certain modèle d'exclusion Markovien appelé "processus d'annihilation asymétrique'' a été étudié par Ayyer et Mallick. Dans ce document, ils ont donné une conjecture précise pour les valeurs propres (avec les multiplicités) de la matrice stochastique. Ils ont en outre supposé que, pour chaque valeur propre, correspond un seul vecteur propre. Nous prouvons la première de ces conjectures en généralisant la matrice originale de Markov par l'introduction de paramètres supplémentaires, calculant explicitement ses valeurs propres, et en montrant que la nouvelle matrice se réduit à l'originale par une spécialisation appropriée. En outre, nous présentons un calcul de la fonction de partition dans le modèle généralisé, ce qui réduit également à celle obtenue par Ayyer et Mallick dans le modèle original.

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Statistical Thermodynamics of Economic Systems

Journal of Thermodynamics ◽

10.1155/2011/676495 ◽

2011 ◽

Vol 2011 ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

Hernando Quevedo ◽

María N. Quevedo

Keyword(s):

Phase Transitions ◽

Partition Function ◽

Probability Distribution ◽

Statistical Physics ◽

Mean Value ◽

Economic Systems ◽

Macroeconomic Variables ◽

Multiplicative Systems ◽

The Mean ◽

Linear And Nonlinear

We formulate the thermodynamics of economic systems in terms of an arbitrary probability distribution for a conserved economic quantity. As in statistical physics, thermodynamic macroeconomic variables emerge as the mean value of microeconomic variables, and their determination is reduced to the computation of the partition function, starting from an arbitrary function. Explicit hypothetical examples are given which include linear and nonlinear economic systems as well as multiplicative systems such as those dominated by a Pareto law distribution. It is shown that the macroeconomic variables can be drastically changed by choosing the microeconomic variables in an appropriate manner. We propose to use the formalism of phase transitions to study severe changes of macroeconomic variables.

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