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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Can we discuss about setting partition function of 1:1 salt electrolyte?

I have 1:1 salt whose density is described by $\rho(r) = \sum\limits_{i=+,-}\sum\limits_{j=1}^{N_{i}}q_{i}\delta(r-r_{ij})$, where $i$ denotes ion species such as +, - and $j$ represents number of $j$ ...
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AdS/CFT and entropy

I would like to calculate the entropy of a CFT on a circle, and I know the partition function of the disk. Can I use AdS/CFT: $$ Z_{CFT} = Z_{\text{bulk}}$$ To deduce : $$S_{CFT} = (1-\beta \partial_\...
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Partition function with two temperatures

I would like to study the thermodynamics of a partition function depending on 2 temperatures: $$Z \ (\beta_1,\beta_2)= \sum_{g=0}^\infty e^{-2g. S_0} \int_0^\infty dl_1 l_1. dl_2 l_2 V_{g,2}( l_1, ...
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Problem with different expressions of functional determinant

This question is a follow-up of my previous one, after having done some calculations. In this previous question I used a minimal example of my problem with $\det(\Delta) = \det(\partial^2+A(x))$, but ...
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Functional determinant: linking Series, Heat-Kernel and Zeta function

I would like to express a functional determinant as a series of diagrams, using the zeta function renormalization applied to the heat-kernel method, but I don't know if it's possible. Let me explain: ...
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Operator insertions vs boundary conditions in AdS/CFT

This question is motivated by AdS/CFT, but really it's just about AdS quantum gravity. Consider quantum gravity in asymptotically AdS spacetime. For simplicity, assume there are no matter fields: the ...
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When can you express the Helmholtz energy as a sum of terms?

In both excess Gibbs/Helmholtz energy models the total Gibbs/Helmholtz energy is the sum of multiple contributions. For example, in the SAFT equation of state the total Helmholtz energy is $$ A = A^\...
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What are $\mathcal{F}_g$ in string theory?

I was reading an article and came up on $\mathcal{F}_g$. Namely, it was in the following equation, $$\psi_{top} = \exp(\sum_g \mathcal{F}_g)$$ where I believe the $g$ denotes the genus of the topology ...
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Boltzmann distribution and probability of finding the system with specific energy

For sake of simplicity assume classical discrete systems. If we have a system ($\text{S}$) coupled to a reservoir ($\text{R}$), then a microstate of the combined (isolated with fixed energy $E$) ...
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Saddle point contributions to the gravitational path integral

In his lectures on black holes and quantum information, Tom Hartman states that the gravitational path integral can be approximated as $$ Z(\beta) \approx \sum_{g_\text{cl}} e^{-I_E[g_\text{cl}, \phi]}...
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2 State canonical ensemble partition function where number of particles in each state is known

If we're asked to find the partition function for a two state system with energies $ E_1 $ and $ E_2 $ for $ N $ indistinguishable, independent classical particles, then it makes sense to me that the ...
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Path integral in Lattice gauge theory with fixed gauge really the same as without fixing the gauge?

In 1 the question why in lattice gauge theories with gauge group $G$, there was no need for gauge fixing to obtain finite path integrals was answered. Thus observables could be calculated as \begin{...
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Physical interpretation of the asymptotics of partition function in string theory

I would like to understand the physical interpretation of the asymptotic expansion of a partition function. The QCD partition function with gauge group $SU(N)$ as $N$ is large has been shown by Gross ...
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Why is the free energy unitless when taking the thermodynamic limit?

Why is the (Helmhotz) free energy unitless when taking the thermodynamic limit? Given the partition function $Z$ of a (finite size) system, the free energy is given by $F =-kT \log[Z]$, where $k$ is ...
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Derivation of the Canonical Ensemble

One of the common derivations of the canonical ensemble goes as follows: Assume there is a system in the contact with heat reservoir which together form an isolated system. Heat can be exchanged ...
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What is a symmetry of the generating functional, and what is the significance?

I cannot find a definition for a symmetry of the generating functional in Quantum Field Theory: $$ Z[J] = \int \mathrm d \mu \, \exp\left\lbrace i S[J] \right\rbrace \, .$$ I know it's a simple ...
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Partition function for $H=-\vec \mu \cdot \vec B$

The interaction energy between a magnetic dipole $\mu$ and a fixed magnetic field $B$ is $E(\theta)=-B \mu cos\theta $ where $\theta$ in the angle between $B$ and $\mu$. The partition function is $Z=\...
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Confusion with Pathria's derivation of the Canonical Ensemble

I've some problems understanding Pathria's derivation of the canonical ensemble and the probability of a system being in a certain state. According to Pathria ( section 3.2 ), we consider an ensemble ...
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Issue with path integrals for the partition function

I was going through Kapusta and Gale "Finite temperature Field theory" In chap 2, Eq. 2.24, they need to do the path integral $$Z = Lim_{N-> \infty} \left (\prod_{i=1}^{N} \int_{-\infty}^{...
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Generalized partition function with angular momentum

I'm studying statistical mechanics, and I found this problem in my study material: Suppose you have a gas consisting of N identical non-interacting atoms in a harmonic trap. Consider its Hamiltonian ...
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The Partition Function of $0$-Dimensional $\phi^{4}$ Theory

My question is related with this question. Several years ago, I posted an answer to the question, and the author of the reference removed the link permanently, now I have no clue what's going on. In ...
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Why does expressing the Faddeev-Popov determinant as this lead to such problems?

Background In the following, I am interested in the Schwinger function associated with the gluon propagator when one considers the Gribov no-pole condition in the partition function. Defining $\nabla^{...
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Quantum versus Classical Partition Function

I am confused as to which partition functions are classical and which are quantum. I am interested in the canonical ensemble. Several places I have seen the "classical" partition function as:...
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Ensemble associated with a system of Photons

In the book "Fundamentals of Statistical and Thermal Physics" by F. Reif it is said that : Photon Statistics is a special case of Bose-Einstein Statistics with no restriction on the total ...
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Should vibrational internal energy of a diatomic gas be considered when calculating below the characteristic temperature $\theta_{vib}$?

(this is a confusion raised from an exercise problem in Carter's Book:) For a diatomic gas, (take $N_2$ for example, which has $\theta_{vib}=3352k$), at room temperature: $\theta_{rot} \ll T=298k \ll ...
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Is it OK to do this manipulation at the partition function level? (auxiliary fields in quadratic gravity)

Background I am working with the following action in the Euclidean signature ($C^2$ is the Weyl quadratic term): \begin{equation} S_B = -\frac{1}{2\kappa^2}\int d^4x\sqrt{g}\left(2\Lambda_C+\zeta R-\...
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String partition function

Hey I am a bit confused because when reading physics papers I encounter the free energy as a generating function of topological invariants as integrals over certain compactified moduli spaces. For ...
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Ideal Fermi gas close to $T=0 \rm K$

I am reading about the behaviour of ideal Fermi gas close to $T=0K$ from Kardar's Statistical Mechanics. In the paragraph which I have highlighted, we have the inyegral representation of $f_m^{-}(z)$....
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High temperature limit of ideal quantum gas

I am reading ideal quantum gas from Kardar's Statistical Mechanics. $VII.35$ is the representation of pressure, number density and energy density in the form of $f_m^{\eta}$. $z=e^{\beta\mu}$ where $\...
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Quantum partition of two integer spins

I am trying to write the partition function for a simple system made of two interacting bosons of spin $S=1$ having the interaction Hamiltonian $$ \hat{H} = \vec{S}_1 \cdot \vec{S}_2 $$ I think I ...
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Partition function for the indistinguishable particles using symmetrization of states

In the derivation of the partition function for the the N particle ideal gas, the factor of $\frac{1}{N!}$ does not come naturally. We have to go for symmetrized and asymmetrized state. So, to derive ...
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How the Zero-point Energy of the System containing 2 Fermions in 3 Micro-Energy States is 1?

If we distribute 2 Fermions $\mathrm{(A,A)}$ in 3 Micro-Energy States (0,$\epsilon$,$2\epsilon$), the confirmation is given by : $$ \begin{array}{|c|c|c|c|c|} \hline 0 & \varepsilon & 2 \...
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Clarification regarding the terminology of Microstates

I would like to understand how microstates are defined or used in physics. Are microstates suppose to only mean eigenvalues of a given observable (or a generator of symmetry)? The reason for my ...
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Free particle in momentum representation

In Greiner, density operator for a free particle has been calculated in momentum basis. They consider a large box of vilume $V=L^3$ and periodic boundary condition. $$\phi_\vec{k}(\vec r)=\frac{1}{\...
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Derivation of partition function for $N$ identical quantum harmonic oscillators

What is the partition function $$\mathcal Z^{(N)}_\beta(H) : =\mathrm{Tr}\exp(-\beta H) \tag{Z} $$ $\left(\beta >0\right)$ for a system of $N$ indistinguishable and non-interacting bosons (e.g. ...
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Computing the partition funciton of 2 identical particles in a harmonic oscillator

Say I have two identical (fermionic) non-interacting particles in a 1D harmonic oscillator. I would like to compute the entropy of the system as the temperature $T$ varies, for which I need the ...
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Partition Function quantum identical particles

In Pathria and Beale Statistical Mechanics section 5.5, the book tries to compute the Partition function of a system of noninteracting, indistinguishable particles confined to a cubical box of volume $...
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Evaluation of thermal average with path integral

I want to evaluate the thermal average $$<\hat{\phi}(0)\hat{\phi}(0)\hat{\phi}(x)\hat{\phi}(x)>$$ with the path integral. $\phi$ is a real scalar field. In general: $$<\hat{\phi}(0)\hat{\phi}(...
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Sums in statistical mechanics

I am evaluating the partition function of a system of particles and incurred in sums like $$ S(a)=\sum_{k=0}^\infty (2k+1)^\frac{\kappa}{2}e^{-(2k+1)a} $$ being $\kappa\in\mathbb{Z}$ and $a=m_0\beta$ ...
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Eigenvalue of transfer matrix using Monte Carlo in 2D Ising model

Montecarlo is an algorithm capable of numerical estimation of any quantity which can be written as the average of a state function like, for example, the magnetization or the internal energy in the 2D ...
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Proving Relationship between Statistical Weight $Ω$ and Partition Function $Z$

How can I prove that $$\frac{\partial(T \ln Z)}{\partial T}=\ln \Omega,$$ without using the relation $S=k\ln Ω$? where $Z$ is the Partition Function, $T$ is the Absolute Temperature, $Ω$ is the ...
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How to Derive the 0 Temperature Action of Bose-Hubbard System?

I am currently reading chapter 8 of the textbook A Modern Approach to Critical Phenomena. This chapter deals with the Bose Hubbard Model and is filled with many equations that aid the analysis of this ...
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Partition function for bosons with path integral

In this book the partition function for bosons is defined in eq. 2.17 as: $$Z=\mathrm{Tr}[e^{-\beta (H-\mu_i N_i)}]=\sum_a\int d\phi_a\langle\phi_a|e^{-\beta(H-\mu_i N_i)}|\phi_a\rangle$$ The ...
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Chemical Potential of an electron or positron in a pair production reaction

In the Thorne and Blandford Modern Classical Physics text there is an exercise that walks you through deriving the density-temperature limit for pair production in plasmas. To do this they consider a ...
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Ratio of particles in energy states

For a system of N weakly interacting distinct particles distributed in 2 energy states and degeneracies given by $\epsilon_{1} = O(g_{1} = I)$ $\epsilon_{2} = 2 \epsilon (g_{2} = 4)$ where $epsilon = ...
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Partition function for weakly interacting gases

I'm studying the Susskind lecture on statistical mechanics. A potential energy of pairwise interactions has been defined: $$\sum_{n>m}U(|x_n -x_m|)$$ We want to calculate the partition function, ...
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Partition Function of a Hydrogen Atom - How it is derived?

It is well-known that the Partition Function of Hydrogen Atom diverges if we calculate in naive manner. And I could find the partition function named Brillouin-Planck-Larkin partition function which ...
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Correlation Function and Generating Functional in QED

Peskin and Schroeder (1995, p.82 and p.292) define the two-point correlation function of a $\phi^4$ theory as $$\langle \Omega|T\{\phi(x)\phi(y)\}|\Omega\rangle\tag{4.10}$$ and the generating ...
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Geometric Entropy & the Partition Function for Conical Deficits

In their paper, Callan and Wilczek claim to derive from the thermal entropy $$S_\text{thermal} = -\left(\beta\frac{\partial}{\partial\beta}-1\right)\ln(\mathcal{Z})$$ a geometric entropy which is ...
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Partition function of a double-stranded dna molecule [closed]

I have a doubled-stranded dna molecule. The molecule has N links, each of which can be one of two states: a closed state and an open state with energy $\epsilon$. A link can be open only if the link ...

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