# The Analytical Expressions for a Finite-Size 2D Ising Model

## TL;DR

This paper derives and validates analytical formulas for the thermodynamic properties of a finite-size 2D Ising model, revealing how finite size affects critical phenomena like heat capacity and critical temperature.

## Contribution

It generalizes Onsager's solution to finite lattices and provides experimentally validated analytical expressions for key thermodynamic quantities.

## Key findings

- Heat capacity at critical point grows logarithmically with system size
- Finite system size limits the precision of critical temperature determination
- Analytical expressions match numerical results for finite 2D Ising lattices

## Abstract

Numerical methods are used to examine the thermodynamic characteristics of the two-dimensional Ising model as a function of the number of spins N. Onsager's solution is generalized to a finite-size lattice, and experimentally validated analytical expressions for the free energy and its derivatives are computed. The heat capacity at the critical point is shown to grow logarithmically with N. Due to the finite extent of the system the critical temperature can only be determined to some accuracy.

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Source: https://tomesphere.com/paper/1706.02541