Questions tagged [partition-function]

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$N$ particles in a box [closed]

I am writing to ask about the problem: $N$ particles in a box.(particles are treated as ideal gas) As you know, $N$ particles will enter the N boxes one by one. First, I entitle the $N$ boxes are ...
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25 views

Partition function for two systems

The goal. Calculate the partition function of the following systems: A gas of $N$ non-interacting distinguishable particles with non-degenerate energy levels $E_0=0$ and $E_1=\epsilon$; A chain of $N$...
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57 views

How can I show that $1/N$ expansion for large $N$ matrix models have a string theoretical perturbation expansion?

While surfing through some further reading suggestions on string theory, I stumbled upon this slide from a talk by Nathan Seiberg. I wanted to derive the main argument by applying a perturbation ...
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65 views

Does Boltzmann's distribution require unbounded energy levels?

How should we interpret Boltzmann's distribution when the set $\{ E_1, E_2, \cdots, E_k \}$ (in increasing order) of energy levels is a finite set? In that case, the expected energy cannot exceed $...
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Statistical mechanics: Island model (WSME model) of protein folding

I have to find the partition function of a system described by the Island Model but instead of having only two species, I have three of which two are disordered that do not contribute to energy. How ...
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64 views

Is $Z=e^{-\beta F}$ as general as $Z=\sum_{Ei}\Omega(E_i)e^{-\beta E_i}$?

I'm having some trouble understanding the formula: $Z=e^{-\beta F}$ which is sometimes used to define the partition function. From what I understand, this $Z$, is the partition function but taken at ...
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47 views

Path integrals on classical statistical mechanics

I'm learning a little bit about path integrals by myself lately and notice something quick curious. So far, I've learned that path integrals have many applications in physics, including quantum ...
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Statistical mechanics: Canonical Partition Function for a Polymer in a lattice

First time poster here so I hope I'm not missing anything. The question "Consider a simple, lattice model for a polymer. In this model, the polymer sits on a square lattice, and at every lattice ...
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1answer
60 views

Why is this the Helmholtz Free Energy for the Onsager Ising Model?

I'm reading through Kerson Huang's presentation of the Onsager solution. We end up determining that the natural log of the partition function is $$\ln Z = \frac{1}{2}\ln (\frac{2 \cosh^2(2 \beta \...
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1answer
47 views

Use of a geometric series sum in derivation of Bose-Einstein distribution

In the following Wiki derivation of the Bose-Einstein distribution, a geometric sum is used to make the following step $$ \sum_{n=0}^\infty\left (\exp \left (\frac{\mu -\epsilon}{k_B T}\right)\right)^...
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Summing over Adjacency Matrices in Partition Function

I'm currently reading a paper (abstract here) on the statistical mechanics of Random Geometric Graphs, and they start with the statistical mechanics of hidden variable graphs. They've taken $a_{ij}$ ...
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52 views

Partition function as a function of inverse temperature

In statistical mechanics, I'm trying to gain intuition as to how to get an upper bound on an equilibrium expectation value of some observable $F$ in terms of the non-interacting expectation value. Let'...
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What are the good references on theory and applications of path integrals and partition functions in theoretical high energy physics? [duplicate]

I have recently been involved with many papers (related to black hole information paradox, etc) in which the concepts of (Euclidean) path integral and partition function are used extensively. Of ...
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About variational methods, renormalization and $a$, $c$-theorems

Variational approximation Variational methods are an important technique, frequently applied for the approximation of complicated probability distributions, with the applications in statistical ...
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Hypothetical sources of partition function of a quantum theory of gravity

Let's assume there exists a partition function $Z[J_i]$ for a theory of quantum gravity, with a family of sources $(J_i)_i$. Given the Palatini action in 3+1 dimensions: \begin{equation} S_P[e,\omega]=...
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1answer
94 views

What is value of critical temperature?

What is the value of critical temperature in the 2D classical ising model? My Understanding Suppose one can write the partition function for the 2D classical ising model in high-temperature expansion ...
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Partition function for a system in local thermal equilibrium

For a system in equilibrium, the partition function is standard. But if the system is in local thermal equilibrium but stationary (i.e. zero or negligible time variation), but the temperature varies ...
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116 views

Estimating partition function using Montecarlo methods

While working on a completely unrelated quantum computing problem, I ran into a quantity that can be mapped to a partition function of spins on a triangular lattice. It is not quite an Ising model, ...
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50 views

Should the same amount of heat increase entropy less in larger systems?

Suppose I have some simple isolated system, so the entropy is given (according to the Wikipedia page on Hemholtz free energy) by $$ S = k\log Z +\frac{U}{T}+c$$ where $Z$ is the partition function, $U$...
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220 views

Is the fully connected Potts model exactly solvable?

Suppose that we have "spins" $\sigma_1,\dots,\sigma_N$, with $\sigma_i\in\{1,\dots,q\}$, for $i=1,\dots,N$, and that our Hamiltonian is $$ H = -\frac{J}{N} \sum_\stackrel{i,j=1}{i\ne j}^N \...
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87 views

Perturbative expansion and self-contractions in functional integral

Consider a one-dimensional integral $$I(g)=\int dx\, e^{-x^2-gx^4}$$ One can formally expand it perturbatively order by order in $g$ so that $$I(g)=\left<1\right>-g\left<x^4\right>+\frac{g^...
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39 views

Partition function of quadrupole? [closed]

I would like to compute the partition function Z for the quadrupole moment, but it is not as easy as the one for the dipole. Could anyone help please?
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1- Loop partition function (Ramond-sector)

When calculating a part of the trace for the partition function of the fermionic Ramond-sector in light-cone coordinates, I'd like to understand how we get to the result $\left(\frac{\theta \left[1/2;...
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194 views

One-loop effective action for scalar field on the curved background in large potential

I hope to compute a functional integral $Z=\int \mathcal{D}\phi\,\, e^{-S[\phi]}$ with an action $$S[\phi]=\int d^2x \sqrt{g}\Big((\nabla \phi)^2+\frac{1}{\lambda}M^2(x) \phi^2\Big)$$ The scalar field ...
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85 views

Partition function in quantum field theory

Why does the partition function include current term in free scalar field $$Z[J] = \int \mathcal{D}\phi \, e^{i \left(S[\phi] + \int d^4x \,J(x) \phi(x) \right)}~$$
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Partition function for 4 spins

I was reading some notes by John Chalker on order by disorder and encountered a classical spins partition function calculation. I could not follow the integration, i.e. obtaining eqn. (1.7) from (1.5) ...
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30 views

A closer look on the derivation of the susceptibility in the Ising-Model

The susceptibility $\chi$ can be defined as $$ \chi = \frac{\partial \langle M \rangle}{\partial H}, \tag{1}$$ where $\langle M \rangle$ is defined as the average magnetization and thus can be written ...
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51 views

What is the entropy and/or equation of state of a partition function such as $Z=\int D\phi \exp (i S[\phi]/\hbar)$?

At this link https://en.wikipedia.org/wiki/Partition_function_(mathematics), it is claimed that the following partition function: $$ Z=\int D\phi \exp (-\beta H[\phi]) \tag{1} $$ is a consequence of ...
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How does anomaly inflow work in terms of the eta invariant?

I'm trying to understand the non-perturbative picture of anomaly inflow, mainly following these two articles by Witten and Yonekura: [1] - https://arxiv.org/pdf/1909.08775.pdf , [2] - https://arxiv....
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49 views

About the various ensembles in Thermodynamics

The properties of a system in thermodynamical equilibrium are described by a partition function: $$ \mathcal{Z} = \text{Tr} \ e^{-\beta E} = \sum_n e^{-\beta E_n} $$ This defines so called canonical ...
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48 views

Where does the radius integral go in the partitition function?

I am confused about a given solution for the following exercise: A thermodynamic system consists of N atoms in the Volume V. Every atom has a magnetic moment. The hamiltonian can be written as a sum ...
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Canonical ensemble: Why do I lose dependency on the number of particles N here?

I have a problem understanding the solution of an exercise that deals with a gas in the framework of the canonical ensemble. Because I'm not a native english speaker some sentences might sound a bit ...
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1answer
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How are the saddle-point equations derived in the single random matrix model?

In this question, I'm referring to a specific step in https://arxiv.org/abs/hep-th/9306153. I want to reproduce equation (2.4) on page 15. I think I lack the experience required for dealing with ...
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27 views

Transfer matrix for Potts models

The interaction energy for Potts model (on an dicrete interval with periodic boundary condition, i.e. a $\mathbb{Z}_N$- model) is of the form $$E(\{\sigma\})=-\sum\limits_{\langle n,m\rangle} \delta_{\...
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50 views

Fermionic Harmonic Oscillator Partition Function

I am reading Nakahara Geometry, Topology, and Physics. In the section on fermionic harmonic oscillator, after some math, the partition function is given by $$\begin{aligned} Z(\beta) &=\mathrm{e}^{...
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149 views

Chemical potential in canonical partition function

I'm a bit confused on the interpretation of the chemical potential in a canonical ensemble (a system which can only exchange energy with a reservoir but not particles). Here is what I think I know: ...
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24 views

How to deal with integral operators in the action, in the path integral of a field theory?

One could imagine adding to the free action of a scalar field theory some non-local operators given as integrals over the base manifold (or over the boundary) of some smooth function of the scalar ...
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47 views

Equation of state of a path integral

How does one take the equation of state of a path integral? In "discrete" statistical physics, one has this partition function: $$ Z=\sum_{i}\exp(-\beta E[i]) $$ And the equation of state is the ...
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61 views

Path integral as a partition function (math)

I am reading the following Wikipedia page, but I am skeptical about what I am reading (it sounds too good to be true). Specifically, I am looking at the passage which states: The number of variables $...
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25 views

Internal Partition Function and Formation Energy

I'm currently reading about internal partition function at: ...
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1answer
85 views

Is estimation partition functions without resorting to markov chain monte carlo still an open question?

I was told estimation of partition functions without resorting to MCMC was still an open question in physics about a year and a half ago. An example is that say you have some physical model that ...
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Where does the factor of $1/V^N$ come from in the classical non-ideal partition function?

I'm reading about classical non-ideal gases from this source. On page 5, they say that the non-ideal partition function is a product of the ideal partition function and the non-ideal partition ...
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1answer
20 views

Thermal expectation from Gauss-Integral partition function

Why is it that, given a partition function $Z$ which takes the form of a Gaussian integral, such equalities hold for the corresponding thermal expectation values, in the example below for the ...
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1answer
140 views

Log of partition function

I know that in statistical mechanics, the partition function of a system of non-interacting, distinguishable particles is of the form $$ Z(T,V,\mu) = \prod_i Z_i(T,V,\mu) $$ where $Z_i$ is the ...
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322 views

What do physicists mean by solving the Ising model?

To me, an Ising model is a setting of discrete objects, that have attributes (spins) that contribute to energy based on interactions with nearby objects. With the energy function (Hamiltonian) written ...
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Asymptotics for two point function of the 1D XY model

Is it correct that in the infinite volume at inverse temperature $\beta$, the two-point function of the classical XY chain magnet behaves like $$ \mathbb{R}\ni x\mapsto\exp(-\frac{x}{4\beta})$$? I am ...
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Is there a statistical physics model derivated from a quantum field theory?

Is there a way to start from a quantum field theory describing some type(s) of particle(s) to obtain predictions about the statistical collective behaviour of a great number of this particles? In ...
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86 views

Fixed Point of RG Flow of $\phi^3$ theory in 6 dimensions

I was calculating RG Flow equations for $\phi^3$ theory in 6 dimensions. The partition function and lagrangian are given below, $$Z = \int D\phi\ e^{-\int^{\Lambda} d^6x \mathcal{L}[\phi] }$$ $$\...
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1answer
93 views

Partition Function for a Ideal Gas - Statistical Mechanics

While I was studying statistical mechanics, I saw this in the book that I'm following: We can divide the partition function into a product, $$ \zeta = \zeta_\text{trans}\zeta_\text{int} $$ where $\...
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53 views

Measure in the Fourier Representation of the Coherent States Path Integral

The problem I have arises in the context of condensed matter physics. I am largely following chapter 4 about functional integration in the book by Altland and Simon. Consider the coherent states path ...

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