# Questions tagged [partition-function]

The partition function describes the statistical properties of a system in thermodynamic equilibrium, and is constructed to represent a particular statistical ensemble (microcanonical, macrocanonical, grand canonical).

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### Pressure of a gas on the inside walls of a cylinder canonical ensemble

First post on here. I need some clarification on this topic. I have an ideal gas rotating with angular velocity $\omega$ in a cylinder of radius $R$ and lenght $L$. Working with the canonical ensemble,...
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### Interpretation of the ideal gas single particle partition function

When considering an ideal gas composed of identical particles, in the semi-classical case, one can find that the single particle partition function is equal to: $$Z(1,T,V) = \frac{V}{\Lambda^3},$$ ...
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### Product of two strongly interacting canonical and grand-canonical ensembles

I am running hybrid MC-MD (Monte-Carlo and Molecular Dynamics) simulations. To put it simply, a nanocluster which atoms belong to a subset $A$ is sampling the canonical NVT ensemble via Newton's laws (...
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### Computational complexity of approximating partition functions

I would like to understand the computational complexity of approximating the partition function of 2D Ising model with complex external magnetic field and complex couplings for the following cases: ...
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### Hubbard-Stratonovich (HS) transform (or similar) for higher order-interactions

I have a question about a generalization of the Hubbard-Stratonovich (HS) transformation to decouple two-body interactions. When dealing with Hamiltonians of the kind H = -\sum_{a}...
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### Degeneracy of states in an ensemble of $N$ harmonic oscillators in $d$ dimensions

In the canonical ensemble consisting of $N$ independent harmonic oscillators in $d$ spatial dimensions one has to evaluate sums like $$\sum_n \ldots e^{-\beta E_n} = \sum_E g_E \ldots e^{-\beta E}$$ ...
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### Calculation of canonical partition function for fermion system with degenerate energy levels

I'm having trouble in visualising the generalized version of the question asked here. We have a system with levels whose energies are $0, \epsilon, 2 \epsilon, ..., n\epsilon$, and the number of ...
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### Yang-Lee circle theorem and the distribution of zeros

I am reading a note on Yang-Lee's zeros. The note first focuses on the grand canonical partition function $\Xi(T, z)$, where $z$ is the complex fugacity conjugate to the number of particles. If the ...
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### Making sense of stationary phase method for the path integral

I am trying to understand this paper/set of notes. I have already seen the following related question: Does the stationary phase approximation equal the tree-level term? but had some trouble following ...
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### Partition function of Hydrogen atoms problem

I know there are several questions asking this problem, but I found this problem has not been solved yet to me. I will repeat the problem and state my view. Consider the statistical mechanics of a ...
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### Interpreting generating functional as sum of all diagrams

The generating functional is defined as: $$Z[J] = \int \mathcal{D}[\phi] \exp\Big[\frac{i}{\hbar}\int d^4x [\mathcal{L} + J(x)\phi(x)]\Big].$$ I know this object is used as a tool to generate ...
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### Is average total energy of two objects is the sum of their individual average energies?

In the context of Boltzmann's distribution, Schroeder states that an average is defined as $$\bar{x}=\frac1Z\sum_sx(s)e^{-\beta E(s)}$$ Where $\beta=1/kT$ and E(s) is the energy corresponding to the ...
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### Homogeneity Restrictions on the Distribution of states in Thermodynamic Systems

The expected energy in the canonical ensemble is given by \begin{split} \langle E \rangle &= \frac{\displaystyle\sum_{i=1} E_i e^{-\beta E_i}}{\displaystyle\sum_{i=1} e^{-\beta ...
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### Examples of Path integral $\neq$ Partition function?

Are there any systems we know of whose partition function is not simply Wick rotation of the path integral? Does anyone know of any examples?
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### Partition function with Hamiltonian depending on parameter

Given a Hamiltonain $H(p,q)$, I know that the classical partition function for a single particle is given by an integral over the phase space $$Z_1 = \frac{1}{h^3} \int e^{-\beta H(p,q)} d^3pd^3q$$ ...
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### Partition function for a $SO(3, 1)$ invariant "Hamiltonian"

Suppose, I look at the $SO(3, 1)$ generalization of $H = \frac{p^2}{2m}$, i.e. $$H = \lambda P^{\mu}P_{\mu}$$ where $P^{\mu}P_{\mu}$ is a $SO(3, 1)$ invariant object and $\lambda$ is some dimensionful ...
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