Questions tagged [partition-function]

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Supersymmetric Quantum Mechanics and the Localization Theorem

I am working through Tachikawa's review on instanton counting arXiv:1412/7121, and in his treatment of Atiyah's localization theorem (see section 3.1.3), he mentions the following equations: $$\text{...
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40 views

Calculating grand canonical partition function in QFT

In thermal field theory, if one has a conserved current, it's convenient to work with the grand-canonical partition function $$Z = \exp \left( H - \mu Q \right) ,$$ where $H$ is the hamiltonian ...
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3answers
98 views

What is free energy in the context of a quantum field theory?

I was reading the papers Large $N$ behavior of mass deformed ABJM theory and New 3D ${\cal N}=2$ SCFT's with $N^{3/2}$ scaling. These papers talk about the free energy in the context of quantum field ...
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Why is a grand partition function defined as the sum of total energies?

In Eq. 22.19, why is this partition function defined as not the sum of internal energy, $U_i$, but the sum of total energy, $E_i$?
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31 views

Taking the derivative of the partition function describing a scalar QFT w.r.t. a constant

Premise: I know this question would be better suited to MathSE, but since it strongly concerns QFT, I'm confident I'll find a more exhaustive answer here. Consider a generic partition function $\...
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27 views

The strict definition of partition function in continuous system

I am confused about the definition of the partition function. From the class of statistical thermodynamics, I've learned that the partition function for a system with continuous energy can be ...
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34 views

Correlation function of partition functions

In this paper the authors study the correlation function of partition functions defined by $$\langle Z(\beta_1) \ldots Z(\beta_n) \rangle = \frac{1}{\mathcal{Z}}\int \mathrm{d}H \, \mathrm{e}^{-L \, \...
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1answer
26 views

Partition Function when the Energy States are Both Discrete and Continuous

Normally for statistical mechanics (in this example I will be only refering to the canonical formalism to keep things simple) we generally have a system that we solved the equations of motion for and ...
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1answer
56 views

Partition function of Hamiltonian without momentum dependence

Considering an ideal gas of $N$ diatomic molecules with dipole momenta $\vec{p}$. Given the Hamiltonian of one molecule $$ H = -\vec{p} \cdot \vec{E} = -pE\cos(\theta)$$ when calculating the partition ...
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41 views

Mean value vs ensemble averages?

When computing the mean magnetic moment of a system with Hamiltonian $$ \mathcal{H} = -\mu \vec{H} \sum_{i=1}^{N} \cos{\theta_i} $$ and external field $\vec{H}=H\hat{e}_z$ we first evaluate the ...
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1answer
42 views

Help in calculating Partition function and choosing the ensemble

I've been given this situation "A surface contains $N$ identical atoms in a fixed position. Every atom can occupy one of two states with energies $E_1$ or $E_2$ and the temperature is $T$." For ...
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26 views

What's the partition function and action of thermofield double state (TFD)?

We know that TFD state is consisted of two CFT at each side, and their partition functions are $$ Z_{L/R}= {\rm tr}\,e^{-\beta H_{L/R}}=\int e^{-S_E}\,\,\,, $$ where $S_E$ is just Euclidean action of ...
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30 views

Average magnetisation in the Ising Model

The Ising Model has energy given by $$ E=-B \sum_{i} s_{i}-J \sum_{\langle i, j\rangle} s_{i} s_{j} $$ where $\langle i, j\rangle$ indicates that the second sum is over each pair of nearest ...
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30 views

Fisher Information in Statistical Mechanics

I am studying the canonical ensemble and it seems to me there is an analogy between derivatives of the partition function, which can extract energy momenta for the system and Fisher score /information....
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1answer
84 views

Show: $\langle n \vert \psi \rangle \langle \psi \vert n \rangle = \langle -\psi \vert n \rangle \langle n \vert \psi \rangle$ [closed]

The book (Altland and Simons, Condensed Matter Field Theory, Ch. 4.2) I am reading makes use of the identity \begin{equation} \langle n \vert \psi \rangle \langle \psi \vert n \rangle = \langle -\psi \...
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22 views

What is the Spectral Form Factor?

In many papers in Random Matrix Theory [1-3] related to quantum chaos (and, in particular, to the SYK model) they analytically continuate the partition function of the system $Z(\beta)$ into $Z(\beta +...
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21 views

Vanishing partition function [duplicate]

I am currently stuck with the following partition function Let the action be $$S(X, \psi^1, \psi^2) = \frac{1}{2} (\partial h)^2 - \partial^2h\psi^1 \psi^2 ,$$ where $h$ is a real function of the ...
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1answer
65 views

Deriving the path integral for periodic boundary conditions

I'm thinking about path integrals with the Euclidean time formalism, where I have partition function $Z=\operatorname{Tr} e^{-\beta \hat H}$. I'm used to the following derivation of the path integral: ...
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38 views

Relation between the canonical partition function and the phase space volume

In Kerson Huang's Statistical Mechanics (2nd ed.), it is claimed that the phase space volume occupied by the canonical ensemble is the partition function: $$ Q_N (V, T) \equiv \int \frac{dp dq}{N! h^{...
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1answer
94 views

Partition function in renormalization

When studying statistical mechanics, renormalization is understood from attempts to calculate partition function by simplifying. (For example, David Tong's lecture note) While I understand that ...
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1answer
67 views

Partition function for $N$ particles, $e^{-Ei}$ term also scaled by $N$? [closed]

My professor didn't go over partition functions explicitly including $e^{E_i}$ terms in his lecture notes for some reason. Do you have to scale the $e^{-E_i/k_{B}T}$ term by $N$ if you have $N$ ...
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56 views

Weighted histogram analysis method (WHAM) equations

I am struggling in deriving the WHAM equations. Among others, I follow this paper by Kumar et al. In the appendix, in eq. 24 they write the density of states as a weighted sum over the "measured" ...
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1answer
113 views

Partition Function of an Ideal Gas

Which is the correct partition function for an ideal (bosonic) gas at high $T$: 1) Sum over the number of particles in each momentum state: $$ z_{\vec{p}} = 1 + e^{- \varepsilon_{\vec{p}}/T} + ... = ...
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2answers
67 views

Why are degenerate states more likely to be filled at a given temperature?

Consider if we have a simple two-level toy model, where the ground state has energy $E_0 = 0$ and the excited state has energy $E_1 = \epsilon$ and degeneracy $g$. The partition function for this ...
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2answers
73 views

The canonical partition function (StatMech)

I recently started my course on Statistical Mechanics, where I have been introduced to the partition function of the canonical (and grand canonical) ensemble. My problem is that I struggle (a lot) ...
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1answer
36 views

Soft condensed matter: the partition function for cluster formation

I am stuck on understanding the form of the partition function presented in my lectures, for self assembly of clusters from monomeric molecules. If there are $N_T$ molecules that can form clusters of ...
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60 views

Coupled quantum oscillator: Field theory

Consider two masses $m$ connected by a spring with a spring constant $k$. Each mass is also connected to the wall using the same springs. The Hamiltonian is $$ H = \frac{p_1^2 + p_2^2}{2m} + \frac{k}{...
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27 views

Confusion on Carlip's approach to Cardy Formula

I'm reading Carlip's approach to Cardy Formula (https://arxiv.org/abs/hep-th/9806026). He considers the partition function on the torus of modulus $\tau$ to be $$\mathcal{Z}(\tau, \bar{\tau})= \text{...
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1answer
46 views

Path integral: uncoupling via staging variables

I am studying the transformation to staging variables for the calculation of path integrals in quantum mechanics, following the scheme presented in the book of Mark Tuckerman "Statistical Mechanics: ...
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1answer
50 views

Free energy along a reaction coordinate

I've come into this issue when trying to understand biased sampling methods, in particular, umbrella sampling, but I think the question is more general. A recurring argument is that, along a reaction ...
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1answer
49 views

Problem with finding the density of states of an $N$-body system

I am having problems solving a particular problem in my Statistical Mechanics course. We have a system that is composed of $N$ non-interacting particles each of mass $m$. The particles are bound to ...
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69 views

Partition function in the non-interacting limit

Let's consider the partition function $$Z(\lambda)=Tr (e^{-\beta H})=Tr (e^{-\beta (H_1+\lambda H_2)})$$ for a quantum system with the Hamiltonian $H=H_1+\lambda H_2$ where $H_1$ is the free part of ...
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1answer
211 views

Partition function in spherical coordinates

Suppose I write the Hamiltonian/energy of my system in spherical coordinates ($r,\theta,\varphi$) with conjugated momentums($p_r,p_\theta,p_\varphi$). How do I calculate the partition function? If ...
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1answer
118 views

1d Ising model: Transfer matrices

we came across a peculiarity when calculating the partition function of $N$ spins $s_i=\pm1$ with Hamiltonian $$H=-J\sum_{i=1}^Ns_is_{i+1}$$ where we impose periodic boundary conditions such that $s_{...
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47 views

Partition Factor for $n!$ degeneracy

What would be the partition function $Z$ if we have an $n!$ degeneracy factor such that $\sum n!e^{-nE_o/kT}$. Assuming its shown that this diverges for infinite $n$, but lets define an $n_{max} = ...
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1answer
118 views

Taking a trace using a continuous spectrum of eigenstates

This may be a simple question, but I have not been able to find an adequate discussion in any source that quite answers it. In many cases in quantum mechanics, traces are evaluated using the ...
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1answer
88 views

Obtaining an expression for spontaneous magnetization in 1D Ising model with $H=0$ from the beginning

The usual trick to find the spontaneous magnetization for the 1D Ising model is to calculate the partition function $Z$ with the Hamiltonian $$\mathscr{H}=-J\sum\limits_{i}S_iS_{i+1}-H\sum\limits_{i}...
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1answer
84 views

Fixed boundaries in 1D Ising model

What are the differences for solving the one dimensional Ising model for fixed boundaries using the transfer matrix, compared with periodic boundaries? this picture show the solution for periodic ...
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128 views

Question on the temperature dependence of the partition function

Let's just say we're looking at the classical continuous canonical ensemble of a harmonic oscillator, where: $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ and the partition function (...
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1answer
91 views

distinguishable particles' Hamiltonian

Let us consider a classical Hamiltonian of a many body system \begin{equation*} H = \sum_{j=1}^N\frac{p_j^2}{2m}+V(\mathbf q) \end{equation*} and let us pass to quantum dynamics by promoting the ...
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1answer
172 views

Interpretation of the Boltzmann factor and partition function

$$p_i = \frac{ \exp\left(-\frac{\epsilon _i}{k_BT} \right)}{Z} $$ $$ Z= \sum_{i} \exp\left(-\frac{\epsilon _i}{k_BT} \right)$$ A) Is $p_i$ the probability of the system having an energy equal to $\...
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22 views

Free energy of non-interacting bosons in mean field

The result As a mathematician I am currently struggling to understand Tóth's Phase transition in an interacting Bose system. An application of the theory of Ventsel' and Freidlin. The free energy (...
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About Witten's path integral formulation of Jones polynomial

In his landmark paper Quantum field theory and the Jones polynomial, Witten proposed that the Jones polynomial can be obtained by the expectation value of the Wilson loop operators over links in the ...
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2answers
271 views

Why is the partition function an integral over momentum and position?

I am learning statistical mechanics through the series of online lectures from Prof Leonard Susskind, and the partition function derived is $$Z = \sum e^{-\beta E_i} .$$ I understand this to be ...
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1answer
34 views

Evaluating the free energy of the grand canonical partition function

From Yoshioka (eq. 10.5), the grand partition is given by $$ \Xi = \sum_{N=0}^{\infty}\ \int_0^\infty \exp \left[ \frac{[E-\mu N-S_I(E,N)T]}{k_B T} \right] \ .$$ It goes on that the value of the ...
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Diatomic Partition Function

Given the following Hamiltonian: $H = \frac { 1 } { 2 m } \left( \left| \mathbf { p } _ { 1 } \right| ^ { 2 } + \left| \mathbf { p } _ { 2 } \right| ^ { 2 } \right) + \frac { \kappa } { 2 } \left| \...
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1answer
31 views

Expand the partition fct. of a simple harmonic oscillator

I come across a expansion of the partition fct. of a simple harmonic oscillator $q$ as: $$q=x^{-1}(1-\frac{x^2}{24}+...) \tag{1}$$ where $x=h\nu/kT$. It’s easy to get $$q=\frac{e^{-x/2}}{1-e^{-x}}=\...
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1answer
60 views

Where do the color indices come back in $SU(3)$ Yang-Mills Quantization?

Can the partition function of $SU(3)$ (the Generic Partition function for a yang-mills theory found on the linked wiki page below), be split into a sum of 8 functional integrals for each gauge field? ...
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289 views

Sum to an integral in deriving equipartition theorem

I'm reading this derivation of the equipartition theorem for ideal gases. On the second page, it is mentioned that the partition function as a simple sum, $${\displaystyle Z=\sum _{i}e^{-\varepsilon ...
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1answer
144 views

Fluctuation of number of particles in one state in canonical ensemble

I haven't touched statistical physics for a while and am stuck in quite a basic question, and surprisingly, I cannot find any information in the internet that helps me to think it through. What is ...