Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [partition-function]

The tag has no usage guidance.

3
votes
2answers
60 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 (...
1
vote
1answer
73 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 ...
4
votes
1answer
124 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 $\...
0
votes
0answers
11 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 (...
1
vote
0answers
58 views

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 ...
1
vote
2answers
68 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 ...
0
votes
1answer
23 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 ...
1
vote
0answers
14 views

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| \...
0
votes
1answer
29 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}}=\...
-1
votes
0answers
67 views

Partition function of an interacting particles

The Hamiltonian of (𝑁+2) interacting classical particles, that are enclosed in a cube of volume 𝑉 at temperature 𝑇, is given by: $H = \sum_{i=0}^{N+1} \frac{|\vec{P_{i}}|^2}{2m} +\frac{1}{2}mw^2 \...
0
votes
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? ...
4
votes
2answers
233 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 ...
1
vote
1answer
40 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 ...
1
vote
1answer
72 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 ...
3
votes
1answer
160 views

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

In quantum field theory, the partition function of a free scalar is $$\mathcal{Z}=\int\mathcal{D}\phi\exp i\int d^{n}x\frac{1}{2}\left[(\partial_{\mu}\phi)(\partial^{\mu}\phi)-m^{2}\phi^{2}\right]$$ $...
2
votes
2answers
94 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-\...
1
vote
0answers
68 views

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 ...
1
vote
1answer
94 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 ...
0
votes
1answer
54 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{ \...
0
votes
0answers
39 views

Deriving an entropy expression from two other expressions

I would like to show that an equation for entropy is given by $$S = k\frac{\partial}{\partial T}\left[T \ln Z\right].$$ I am given the equations $$S = -k_{B}\sum_{i = 1}^{N} p(n_{i})\ln(p(n_{i}),$$...
3
votes
1answer
81 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 ...
1
vote
1answer
68 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 ...
1
vote
0answers
66 views

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\...
3
votes
0answers
39 views

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 $G_\cap$ ...
0
votes
0answers
50 views

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 ...
1
vote
0answers
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 ...
1
vote
2answers
117 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 ...
1
vote
0answers
57 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 ...
1
vote
1answer
53 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 ...
3
votes
1answer
92 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....
1
vote
0answers
62 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 ...
3
votes
0answers
77 views

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) \...
1
vote
1answer
184 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) \...
0
votes
0answers
26 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. ...
1
vote
1answer
106 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 ...
2
votes
1answer
72 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. ...
2
votes
1answer
157 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 ...
0
votes
1answer
85 views

Semantic distinction between “Partition Function” and “Generating Functional” in QFT?

I am just now learning about these, and I have seen them defined as follows: The generating functional for a set of fields $\phi_i$ is defined by: $$Z[J_i]=\int\mathcal{D}\phi_i e^{i(S[\phi_i]+\int ...
0
votes
0answers
118 views

Issue with calculating free fermionic propagator from partition function

$\newcommand{\D}{\mathcal{D}}$ In section 14.6 of Schwartz's "QFT and the Standard Model (7$\,^{\text{th}}$ printing)" [1], the author calculates the exact free fermionic partition function and, from ...
0
votes
2answers
58 views

Determination of chemical potential

I've written a partition function for a problem $Z_1=e^{\mu+\beta\varepsilon_1}+2e^{\mu+\beta\varepsilon_1}$ and calculated the free energy for N particles $F=-kT\ln{Z_1^N}$. I'd like to get the ...
0
votes
0answers
165 views

Calculating free energy from partition function

So I have $N$ particles and I've determined the partition function for one of them to be $Z_1=\lambda e^{\beta\varepsilon_1} + 2\lambda e^{\beta\varepsilon_2}$. I know the free energy is $A=-kT\log(Z)$...
4
votes
0answers
134 views

Extra $i$ in grand canonical partition function: why the Wick rotation?

Going through my notes I stumbled upon something I can't wrap my head around. I'd like to write the grand canonical partition function for a system of identical charged particles (charge $e$) ...
3
votes
1answer
69 views

Magnetism - Large Spin Limit

In class we were computing the partition function given the following Hamiltonian: $$H = - g\mu_B \sum_{i} \vec{h}.\vec{S_i}$$ where $\vec{h}$ is the external magnetic field, and $\vec{S_i}$ is the ...
1
vote
1answer
142 views

Apparent discrepancy between partition function from density matrix and partition function from counting microstates in finite level system

Consider a quantum two-level system indexed by states $|l\rangle = |0\rangle,|1\rangle$ and energies $\epsilon_l$, where $\epsilon_0 = 0$,$\epsilon_1 = \epsilon$. I throw in 2 bosons into the system ...
0
votes
0answers
259 views

Derivation of partition function

So this follows Schroeder's Intro to Thermal Physics equations 6.1-6.7, but the question isn't book specific. Please let me be clear: I know for a fact I'm wrong. However, it feels like performing ...
0
votes
1answer
73 views

Is it possible to derive the partition function from action?

According to thermal physics, any system, after an infinite amount of time, will obey the Boltzmann distribution. However as each particle follows a trajectory that minimises the action surely it ...
0
votes
0answers
89 views

Equivalence of High and Low temperature expansion of 2D Ising model partition function

It is known that it is possible to find a duality in the 2D Ising model rectangular lattice (the one with coordinates in $\mathbb{Z}^2$). Let the number of pairs of neighbors be $s$ and the total ...
0
votes
0answers
28 views

Does the partition function define probability of being in multiple states?

The partition function is defined as a sum over all microstates $j$ as: $Z=\sum_{j}exp(-\beta E_j)$ or $Z=\int_{-\infty}^{\infty} exp(-\beta E)dE$ if the states are continuous. We can use $Z$ to ...
1
vote
1answer
155 views

Problem understanding Fermi - Dirac distribution function

I am studying Statistical Mechanics by R K Pathria. There the author tries to calculate the partition function of in canonical ensemble as: $$Q_N(V,T) = \sum_{[n_\epsilon]} (e^{-\beta\sum_{\epsilon}...
3
votes
1answer
154 views

Correlation Function of One-Dimensional XY Model

From the Harvard lecture notes XY model: particle-vortex duality by Subir Sachdev, the path-integral of 1D XY-model is given by $$\mathcal{Z}=\int\mathcal{D}\theta\exp{\left\{-\frac{K}{2}\int \!dx~(\...