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Questions tagged [partition-function]

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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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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
29 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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46 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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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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41 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
40 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
39 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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62 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
80 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
52 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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31 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
51 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
79 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
33 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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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
78 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
139 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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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
104 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
24 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
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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
59 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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2answers
250 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
76 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 ...
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1answer
84 views

Why we ignore off-diagonal elements in partition function?

In quantum statistical mechanics, the density operator is $$ \rho = \exp(-\beta H_0)/Z $$ where $$Z = \text{Tr} (\exp(-\beta H_0)) \, .$$ Why do we take the trace over only diagonal elements and ...
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1answer
184 views

Determinant of d'Alembert Operator $\mathop\Box-m^{2}$

In quantum field theory, the partition function of a free scalar is $$\mathcal{Z}=\int\mathcal{D}\phi\exp i\int d^{n}x\frac{1}{2}\left[(\partial_{\mu}\phi)(\partial^{\mu}\phi)-m^{2}\phi^{2}\right]$$ $...
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2answers
124 views

Partition function for an interacting bosonic mode

Let us assume a single bosonic mode, in equilibrium with a reservoir. For a non-interacting Bose gas, the partition function becomes $\mathcal{Z_\text{nonint}}=\sum_{N=0}^\infty e^{-\beta(\epsilon-\...
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Laplace Transform Density of States & Partition function

I am currently going through Pathria's Statistical Mechanics text , and under the Canonical Ensemble description, the author stresses that the partition function of a continuous system is the Laplace ...
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1answer
139 views

Canonical partition function for different systems

As a homework exercise for Advanced Statistical Mechanics I need to derive the canonical partition functions for the following systems: Single component ideal gas on a square lattice Single component ...
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1answer
64 views

Why is potential energy zero in this calculation of partition function?

In section 7.2 of Rief's "Fundamentals of Statistical and Thermal physics". While calculating the partition function for ideal gas he writes: $$ \begin{array}{l} \displaystyle{Z' = \int{ \...
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1answer
158 views

Finding the partition function for a three-level system

I am having difficulty finding the partition function of a system with two particles, each of which can be in any of three states with energies $0, \epsilon, 3\epsilon$. The system is in contact with ...
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1answer
96 views

Partition function of adsorbing molecules

I have the following example: A surface, having $N_0$ adsorption centers, has $N \le N_0$ gas molecules adsorbed on it. Disregarding interactions between the adsorbed molecules. An adsorption center ...
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Majorana Fermion Coherent States

I was wondering if there are coherent states for Majorana operators, so, states that fulfill the relation \begin{align} \hat{\gamma}_A |a,b\rangle &= a |a,b\rangle \\ \hat{\gamma}_B |a,b\...
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Partition function inequality for Gibbs states associated with graphs

Suppose I have two undirected graphs $G_1$ and $G_2$ with the same vertex set $V$ and let $A_1$ and $A_2$ denote their respective adjacency matrices. Define the intersection of the two graphs $G_\cap$ ...
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Entanglement entropy calculation

I have been trying to understand this paper: Susskind. “Black Hole Entropy in Canonical Quantum Gravity and Superstring Theory.” [1402.1128] Long Short-Term Memory Based Recurrent Neural Network ...
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50 views

In Monte Carlo integration for Molecular dynamics simulation, why is a Boltzmann distribution assumed?

In statistical physics, The calculation of partition function for an ensemble takes a Boltzmann's distribution of the Hamiltonian. Similarly, In Monte-Carlo integration of Molecular Dynamics ...
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2answers
127 views

Functional integration for the order parameter in $XY$ model

In the continuum limit the Hamiltonian of the classical XY model is given by, ignoring the inessential constant: $$H=\int d\vec{r}\ (\nabla\theta)^2$$ and the x-component of the order parameter is ...
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59 views

Where does this $dy$ come from?

CONTEXT: Large Deviation Theory Textbook: Perspectives on Statistical Mechanics, Yoshitsugu Oono For i.i.d. stochastic variables $\{X_n\}$, the rate function (or large deviation function) $I(y)$ is ...
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1answer
54 views

mechanical statistics - computaion of partition function

Let us consider an ideal gas of $N$ particles contained in the volume $V$ with unitary spin $\vec S$. In particular, the z-component of the spin is $S^z = -1,0,1$. In an external magnetic field $\vec ...
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1answer
127 views

Exponential form of Boltzmann Distribution

I am trying to understand why the Boltzmann distribution is of the specific form: $$ f(E) = A e^{-E/B} $$ for some constants $A$ and $B$. I am following the discussion here: https://courses.physics....
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68 views

Indexes in the Gaussian functional integral

This is a question spawning from a comment made to my previous question. There I was asking about taking some functional derivative in the effective action of the non-linear sigma model. The comment ...
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How to calculate correlators in a 1D Conformally invariant Matrix Model?

I am working on a 1D Conformally invariant Matrix Model with the following Partition function: $$ Z(g) = \int \mathcal{D}M(t) \exp \left[ -\text{tr}\int dt \left( \frac{1}{2} \dot{M}^2(t)+V(M) \...
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1answer
219 views

Effective action of QED and the partition function

Given the partition function for QED $$ Z= \int \mathcal{D}A_{\mu}\mathcal{D}\Psi \mathcal{D}\bar{\Psi}\, \text{exp}\left(- \frac{i}{4}\int F_{\mu\nu} F^{\mu\nu} + i \int \bar{\Psi} (i {\not} D-m) \...
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32 views

Calculating pressure derived from path integral

I have a little problem in mathematically deriving the thermodynamic potential $\Omega =-\frac{\ln Z}{\beta}$ and pressure $P=\frac{ln Z}{\beta V}$ from partition function derived from path integral. ...
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1answer
147 views

How do I find the pressure of the Einstein's solid?

Einstein's solid is a model of solid in which there are N atoms each of which is assumed to be noninteracting, identical and localized quantum mechanical oscillators. The canonical partition function ...
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1answer
83 views

Is the Gibbs free energy not/less important for canonical ensembles? If so, why?

From canonical ensembles, we find that the Helmholtz free energy $F=U-TS$ is related to the canonical partition function as $$F=U-TS=-k_BT\ln Z$$ where $Z$ is the canonical partition function. ...
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1answer
201 views

Grand canonical partition function of hypothetical particles

I have to calculate the grand canonical partition function of a system of hypothetical particles, wherein each single-particle quantum state can be occupied by up to 3 particles. Obviously this is a ...