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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What is the relation between the partition function from Stat. Mech. And the Path Integral? [duplicate]

Beside the fact that they look identical when you take imaginary time in the path integral formulation. I understand we doing statistics and we are just integrating over all states with a relative ...
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Conditions on the covariance operator in Gaussian Path Integrals

In field theory, one typically encounters integrals of the form: $$ \mathcal{Z}[J] = \int \mathcal{D}[\phi] \exp \left( - \frac{1}{2} \int d^Dx d^Dx' \ \phi(x)A(x,x')\phi(x')+ \int d^Dx \phi(x) J(x)\...
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Expectation value of energy in a canonical ensemble

Suppose we have a canonical ensemble system of $N$ particles, and $k$ single-particle-energy levels labeled by $\epsilon_i$. The energy of the different microstates of the entire system is given by $\...
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Is the total energy of a canonical ensemble system of $N$ particles, with single-particle energy levels given by $\epsilon_i$ fixed?

Is the total energy of a canonical ensemble system of $N$ particles, with single-particle energy levels given by $\epsilon_i$ fixed ? We know the total energy of the system is given by : $$E=\sum_{i} ...
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Interpretation of probability in Statistical Mechanics

In statistical mechanics, in particular the canonical ensemble, the probability of the system to have a particular state is given by : $$P_i=\frac{e^{-\beta E}}{Z}$$ Here $Z$ is the partition function ...
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Confusion regarding the use of partition function

Suppose we have a system filled with $N$ particles. There are $k$ energy levels in this system, labeled by $\epsilon_i$, each with a degeneracy of $g_i$. Let us imagine $n_j$ particles out of these $N$...
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Confusion about using single particle or $N$ particle partition function in Boltzmann probability in canonical ensemble

Suppose we have a canonical ensemble, where $N$ particles have been divided among $\epsilon_i$ energy levels, each with degeneracy $g_i$. The partition function for a single particle is given by : $$...
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Standard Relation in terms of partition function and beta

$$(\Delta E)^2 = \frac{\partial^2 \ln{Z}}{\partial \beta^2} \tag{2.29}$$ Shortly, how can I obtain this relation? I found this relation from Franz Mandl Statistical Physics The following are related ...
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Quantum to classical mapping

I'm having troubles understanding precisely how the mapping from a quantum system to a classical one works. Let's say that I have a quantum system in $d$ dimensions with Hamiltonian $H$ at temperature ...
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Free energy landscape and probability distributions

I couldn't understand the concept of free energy landscape. Usually free energy is defined in the following way. $$F=-k_{B}T\ln Z=-k_{B}T\ln\sum e^{-\beta \epsilon}$$ where $\epsilon=\sum\frac{p_{i}^{...
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Partition function of 3D quantum harmonic oscillator

The following discussions are for isotropic quantum harmonic oscillators which have the energy eigenvalues as follows: $$E=\left(\sum_{i}^{N}n_i+\frac{N}{2}\right)\hbar \omega$$ where $N= $ number of ...
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Entropy-per-site model leading to "shapes" of generating function roots

This question is cross-posted from MathOverflow. Many statistical mechanics systems have well-defined entropy-per-site function $f(x)$ with respect to some control parameter $x$. This question is ...
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Interpretation of "pressure" as the logarithm of the partition function

Consider the Ising model on a subset $\Lambda\subset\mathbb{Z}^d$, with partition function $Z_{\Lambda; \beta , h}$ where $\beta$ is the inverse temperature and $h$ the external magnetic field. The ...
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XY model free energy

For the XY model, we have $$ Z = \int_0^{2\pi} \prod_{i=1}^N d\theta_i \exp(\beta J \sum_{i=1}^N \cos(\theta_i - \theta_{i+1}))$$ and eigenvectors $\vec{v}(\theta)=e^{in\theta} $ and eigenvalues $\...
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Matrix element and Dirac notation

If $$ T= \left[ \begin{array}{cccc} e^{\beta J} & e^{-\beta J} \\ e^{-\beta J} & e^{\beta J} \\ \end{array} \right] $$ and $$Z = \sum_{S_i=\pm 1} ... \sum_{S_N=\pm 1} \exp{\beta J(\vec{...
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CFTs that are not modular invariant

Are there any 2d CFTs that do not have modular invariant partition functions? All the examples that I know of, like the free boson, WZW models, etc. have modular invariant partition functions.
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Partition Function from Dispersion Relation in Molecular Dynamics

I’ve seen there are ways to compute the dispersion relation of a crystal from molecular dynamics. An example of how to do this is discussed in this question: Computing phonon dispersion from molecular ...
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Complete the square for the generating functional of the Dirac field

Quote Peskin page 302 the Dirac generating function was $$Z[\bar \eta ,\eta ]=\int D\bar\psi D\psi\exp[i\int dx^4 (\bar\psi (i\gamma^\mu\partial_\mu -m )\psi+\bar\eta \psi+\bar\psi \eta)]$$ could be ...
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How to write the spatial photon propagator of the generating function $Z[J]$ in QED?

There was a part in the lecture one didn't quite understand. (The charge $e=0$ in this post.) The partition function for massive fields such as scalar fields or the spin-1/2 fields were quite standard....
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In QFT, why are the vacuum partition function and the zero-temperature imaginary-time partition function the same?

When doing thermal field theory, one can start with the definition of the (thermal) partition function $Z = Tr[e^{-\beta H}]$, and inserting a number of completness-relations, we can arrive at (I am ...
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Grand canonical Partition function of fluids

I'm going through the book Lectures on Phase transitions and Renormalization Group by Nigel Goldenfeld. In chapter two section 2.11 (Fluids), the grand partition function describing the fluid is : $\...
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What is the difference between configuration and microstates in the distribution of fermions

Imagine a system of $3$ electrons distributed into $3$ energy levels ($E_1,E_2,E_3)$. I want to know the difference between the total number of configurations of the above system vs the total number ...
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Evaluating Correlation Function for the Classical Ising Chain

I am currently reading the second chapter of Quantum Phase Transitions by Subir Sachdev. I have the first edition of this textbook. The main idea of the second chapter is the following: quantum phase ...
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Normalization of the partition function

I am used to think that multiplying the partition function by a constant is meaningless, and shouldn't change any calculation, as under $Z' = A Z = A \sum_n \exp(-\beta E_n)$ for an arbitrary $A$, the ...
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How to find the partition function of the $1$D Ising model?

I am trying to solve a problem that requires finding a partition function. Question: Consider a one-dimensional Ising model with $N$ spins at very low temperature. Let there be $r$ spin flips with ...
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Do finite sized 1D Hamiltonians have free energies which are analytic everywhere in the complex plane?

It's well known that 1D classical and quantum short-ranged Hamiltonians have free energies which are analytic/holomorphic everywhere as a function of inverse temperature $\beta=1/k_BT$ (see Araki, &...
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The full path integral of a quantum field theory

Suppose if one is able to do a full path integral of a QFT with an action say $S[\phi]$ i.e. $$Z = \int [\mathcal{D}\phi] e^{iS[\phi]}.$$ What can I use $Z$ for? Can I use the $Z$ like the partition ...
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Free massive scalar field partition function in QFT?

Consider the (euclidean) path integral for the free massive scalar field in $d$ dimensions, giving the partition function $$Z_m=\int\mathcal{D}\phi~e^{\int dx^d~\phi(\Delta-m^2)\phi}$$ with Laplace-...
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Grand partition function substate average number of particles

Let's say I have a grand partition function with two states $\epsilon_1$ and $\epsilon_2$: \begin{equation} Z = \sum\limits_i \exp\left[-\beta\left(\epsilon_1 n_1+\epsilon_2 n_2 - \mu n_i\right)\right]...
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Non-factorization of partition functions in AdS/CFT

I have seen this comment a lot in recent papers. This has to do with the AdS/CFT correspondence and Euclidean asymptotically AdS wormholes. So when we calculate the partition function of two $D-1$ ...
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Confinement in 3+1 dimensions from confinement in 2+1 dimensions

It is well known that Yang-Mills theories in $2+1$ dimensions exhibit the color confinement property. This property is characterized by the average of a Wilson loop that is the exponential of a term ...
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Partition function in SYK Model

In SYK model, we have the partition function for $N$-interacting fermions as \begin{equation} z=\int d^{N} \psi \exp \left(\imath^{q / 2} \sum J_{a_{1} a_{2} \ldots a_{q}} \psi_{a_{1} a_{2} \ldots a_{...
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Can Feynman-Kac formula relate to partition function in a rigorous way?

Feynman-Kac formula reads: $$\psi(x,\beta) = \int e^{\int V({\bf{x}}(\tau))d\tau}\psi_{0}({\bf{x}}(\beta))d\mu_{x}({\bf{x}}). \tag{1}\label{1}$$ This is a rigorous formula, defined by means of a ...
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The model of a free particle in a box as an approximation for the translational motion of a particle in a gas

According to the book Introduction to Physical Gas Dynamics (starting from page 120), the translational motion of a particle in a gas in thermal equilibrium may be approximated by a free particle in a ...
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Alternative derivation of NVT partition function

I wanted to use an approach based on the Liouville equation to formulate the partition function of the NVT ensemble from its conservation laws. You can see some more about this approach in https://...
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Fokker-Planck Partition function

Does a multidimensional Fokker-Planck equation always have a stationary solution? Is it posisble to write this solution in a partition-function-liek for to enable the analysis using the methods of ...
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Landau free energy, ising mean field and the "full partition function". Discrepancy between two similar approaches

From what I understand, for example, in the neighbor interactions Ising model, we can write the partition function as: $$Z = \sum_{m}\Omega(m)e^{-\beta E(m)}=\sum_{m}e^{-\beta \tilde F(m)}=e^{-\beta F}...
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Path integral with constraints

First, I must say that I'm not very familiar with the path integral formalism, so maybe I'm missing something very basic. In Section III.A. of this paper, Toms considers a particle in $D$-dimensional ...
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What is meant by a probablity given by $e^{-\text{P.E.}/kT}$ with $\text{P.E.}<0$?

This is from https://www.feynmanlectures.caltech.edu/I_40.html Let us take the case of just two molecules: the $e^{-\text{P.E.}/kT}$ would be the probability of finding them at various mutual ...
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What is the physics meaning of the "trace" (Tr) and how we can calculate it? [duplicate]

What is the physics meaning of the "trace" (Tr) and how we can calculate it? $$ Z=tr\left \{ e^{\left ( -H \right )} \right \} $$ Where Z is the partition function .
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How to deal with path integral in curved space-time for a free scalar field?

Let's say we have a complex scalar field in a curved background whose action is: \begin{equation} S=-\int d^4x \sqrt{-g}\phi^\ast(\square_g+m^2) \phi \end{equation} For some purpose I want to ...
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Canonical ensemble: what if the phase space density is not known?

In canonical ensemble, the probability is defined as \begin{equation} P(E)=\frac{g(E)\exp(-E/T)}{Z}, \end{equation} and the partition function is defined as \begin{equation} Z(T)=\int_0^{\infty}dE\,g(...
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Partition function of a photon gas

I am trying to calculate the partition function of a photon gas. The book I'm currently following is "Thermal Physics by Garg, Bansal, and Ghosh" It does the following: The parition function ...
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How come that $\int \delta(H(p,q)-E)dpdq=\Omega(E)$ not infinity?

In microcanonical ensembles we have (for one particle in 1 dimension) $$\int \delta(H(p,q)-E)dpdq=\Omega(E)$$ I am not convinced and believe that this integral diverges. Take for example a harmonic ...
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Connected part of $S$-matrix generating functional

I am currently studying an article by A.Jevicki et. al. (https://doi.org/10.1103/PhysRevD.37.1485) and I am a little confused. They say that the generating functional of the $S$-matrix is related to ...
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Taking functional derivatives of generating functional

I'm learning how to compute functional derivatives of generating funtionals. Suppose I have the following generating functional: $$Z[J] = \exp\{\int{dy_1 \; dz_1\; J(y_1) \Delta(y_1 - z_1) J(z_1)}\}$$ ...
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How to calculate a TQFT Gaussian path integral from Seiberg's "fun with free field theory"?

In his talk "Fun with Free Field Theory", Seiberg discusses a topological quantum field theory in $d+1$ dimensions with the action $$ S = \frac{n}{2\pi} \int \phi\, \mathrm{d} a \tag{1}$$ ...
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On the transfer matrix formalism

Classical solutions/studies of the one-dimensional and two-dimensional Ising model make use of the transfer matrix. The following is based on Huang's book. Let us consider a two-dimensional Ising ...
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Partition function of fermions with spin

How do you write the partition function of fermions that can have spin 1/2 and spin -1/2, in a system with N states. I know fermions with the same spin cant occupy the same state, but how do I ...
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How to relate these 2 expressions for expressions for 2-point correlation functions?

I am reading the paper https://arxiv.org/abs/1808.07032 "Statistical mechanics of a two-dimensional black hole". They consider some field theory on hyperbolic space $H^2$ with local ...

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