Questions tagged [partition-function]

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Transfer matrix for Potts models

The interaction energy for Potts model (on an dicrete interval with periodic boundary condition, i.e. a $\mathbb{Z}_N$- model) is of the form $$E(\{\sigma\})=-\sum\limits_{\langle n,m\rangle} \delta_{\...
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40 views

Fermionic Harmonic Oscillator Partition Function

I am reading Nakahara Geometry, Topology, and Physics. In the section on fermionic harmonic oscillator, after some math, the partition function is given by $$\begin{aligned} Z(\beta) &=\mathrm{e}^{...
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111 views

Chemical potential in canonical partition function

I'm a bit confused on the interpretation of the chemical potential in a canonical ensemble (a system which can only exchange energy with a reservoir but not particles). Here is what I think I know: ...
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How to deal with integral operators in the action, in the path integral of a field theory?

One could imagine adding to the free action of a scalar field theory some non-local operators given as integrals over the base manifold (or over the boundary) of some smooth function of the scalar ...
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43 views

Equation of state of a path integral

How does one take the equation of state of a path integral? In "discrete" statistical physics, one has this partition function: $$ Z=\sum_{i}\exp(-\beta E[i]) $$ And the equation of state is the ...
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43 views

Path integral as a partition function (math)

I am reading the following Wikipedia page, but I am skeptical about what I am reading (it sounds too good to be true). Specifically, I am looking at the passage which states: The number of variables $...
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Internal Partition Function and Formation Energy

I'm currently reading about internal partition function at: ...
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64 views

Is estimation partition functions without resorting to markov chain monte carlo still an open question?

I was told estimation of partition functions without resorting to MCMC was still an open question in physics about a year and a half ago. An example is that say you have some physical model that ...
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40 views

Where does the factor of $1/V^N$ come from in the classical non-ideal partition function?

I'm reading about classical non-ideal gases from this source. On page 5, they say that the non-ideal partition function is a product of the ideal partition function and the non-ideal partition ...
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20 views

Thermal expectation from Gauss-Integral partition function

Why is it that, given a partition function $Z$ which takes the form of a Gaussian integral, such equalities hold for the corresponding thermal expectation values, in the example below for the ...
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76 views

Log of partition function

I know that in statistical mechanics, the partition function of a system of non-interacting, distinguishable particles is of the form $$ Z(T,V,\mu) = \prod_i Z_i(T,V,\mu) $$ where $Z_i$ is the ...
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What do physicists mean by solving the Ising model?

To me, an Ising model is a setting of discrete objects, that have attributes (spins) that contribute to energy based on interactions with nearby objects. With the energy function (Hamiltonian) written ...
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Asymptotics for two point function of the 1D XY model

Is it correct that in the infinite volume at inverse temperature $\beta$, the two-point function of the classical XY chain magnet behaves like $$ \mathbb{R}\ni x\mapsto\exp(-\frac{x}{4\beta})$$? I am ...
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Is there a statistical physics model derivated from a quantum field theory?

Is there a way to start from a quantum field theory describing some type(s) of particle(s) to obtain predictions about the statistical collective behaviour of a great number of this particles? In ...
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Fixed Point of RG Flow of $\phi^3$ theory in 6 dimensions

I was calculating RG Flow equations for $\phi^3$ theory in 6 dimensions. The partition function and lagrangian are given below, $$Z = \int D\phi\ e^{-\int^{\Lambda} d^6x \mathcal{L}[\phi] }$$ $$\...
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Partition Function for a Ideal Gas - Statistical Mechanics

While I was studying statistical mechanics, I saw this in the book that I'm following: We can divide the partition function into a product, $$ \zeta = \zeta_\text{trans}\zeta_\text{int} $$ where $\...
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Measure in the Fourier Representation of the Coherent States Path Integral

The problem I have arises in the context of condensed matter physics. I am largely following chapter 4 about functional integration in the book by Altland and Simon. Consider the coherent states path ...
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Calculating partition function at the core of the Sun, to use in the SAHA equation

I have to calculate the percentage of ionised Hydrogen in the core of a star with conditions similar to that of the Sun, using the SAHA equation. As part of this I have to calculate the partition ...
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Correlation functions of currents on torus at critical level

In a paper "CORRELATION FUNCTIONS OF CURRENT-ALGEBRA THEORIES ON THE TORUS" by Mathur and Mukhi the torus correlation function of currents are computed. Crucially, they use the relation $$ \sum_a (J^a ...
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Issue when calculating the partition function for a spring

I have a question where I need to write down the partition function of a classical spring with $E = \frac{1}{2}kx^2$. The goal is to show thermodynamically that the force is $-kx$. From $U = -\frac{\...
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19 views

Unconditional probability of for a system microstate in canonical ensemble

In Kardar's 'Statistical Physics of Particles', it is stated that the unconditional probability for a microstate $\mu_S$ of system $S$ (in a canonical ensemble made using a system $S$ and reservoir $R$...
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Partition function for quantum Ising model

I have hamiltonian for fermionic field as $${\cal H}_F=E_0+\int dx[\frac{v}{2}(\Psi^\dagger\frac{\partial \Psi^\dagger}{\partial x}-\Psi\frac{\partial \Psi}{\partial x})+\Delta\Psi^\dagger\Psi]\tag{1}$...
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75 views

The well-defined temperature and 0D Ising model (Ref. Shankar)

I’m reading Shankar’s book Quantum field theory and condensed matter. On page 17, these two bold sentences seem to contradict each other: The system in contact with the heat bath and described by Z ...
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62 views

Partition function of an ideal gas (taking gravity into account)

I am trying to solve the following problem from an old qualifying exam: "Ideal gas in gravitational potential" Consider an ideal gas of N indistinguishable molecules of mass m in a cylindrical ...
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33 views

Statistical Mechanics: Particles on a Sphere

$K$ identical particles of mass $m_0$ are bound to move on a sphere of radius R. The system is at equilibrium at temperature $T_0$. 1 - What's the internal energy ($E$)? 2 - What's the specific heat ...
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110 views

How to connect Green function to propagator?

I know that there has already been many questions related to this question, such as in Differentiating Propagator, Green's function, Correlation function, etc. However, that question mainly ...
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71 views

The volume factor of a partition function

Something that has bugged me for years but I've never taken the time to ask. Given an ideal gas in a box of volume V and N molecules of gas. The states of the systems is all the coordinates and ...
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16 views

Question about the quadratic deviation of energy in the thermodynamic limit

We know that $$ \langle ( E_j - \langle E_j \rangle )^2 \rangle = - \frac{\partial}{\partial \beta} \langle E_j \rangle $$ And in the thermodynamic limit $ \langle E_j \rangle \longrightarrow U $ ...
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33 views

Correct Boltzmann counting for the partition function for diatomic molecule

Would it be reasonable, given the following: to use a factor of $\frac{1}{2^N N!}$ as opposed to the standard $\frac{1}{ N!}$ for the correct Boltzmann counting term in the partition function for the ...
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27 views

How to prove this expression of the specific heat in statistical physics

Where does thes expression come from? $c_v=\pi^2/3\cdot k_B^2\cdot g(E_F)$ where $g(E_F)$ is the density of state
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Relation between grand potential and expected number of particles in an energy state?

During the review of my lecture notes I stumbled upon an equation that gives me some trouble understanding. The big task that motivates the following is to express the entropy $S$ with the expected ...
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Statistical mechanics, partition function, and probability

I would like to confirm if the distribution $\rho$ of natural parameter $\varphi$ written as $\rho$ =$\frac{1}{Z}$$H(\beta)$$e^{(\varphi T(\beta))}$, where $Z=\int H(\beta)$$e^{(\varphi T(\beta))}d\...
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108 views

Partition function of two spin 1/2 particles - Distinguishable or indistinguishable?

Suppose I have some fermions with spin 1/2 on a harmonic potential. Then the energy of each particle is given by: $$ E_i=\hbar\omega(n_{x_i}+n_{y_i}+n_{z_i}+3/2) $$ By definition the partition ...
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Partition function of 2D Ising model on a squared lattice in the canonical ensemble in the low temperature limit

I'm currently working through David Tong's script on statistical mechanics (http://www.damtp.cam.ac.uk/user/tong/statphys/sp.pdf) and came across something that I don't quite understand (page 166). ...
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Distinguishability of classical particles with magnetic moment

Given a system of $N$ identical particles, under external field $\vec{H}$ in fixed volume $V$ and each particle has energy given by $\epsilon_{\vec{p},H}=\dfrac{\vec{p}^2}{2m} + \mu_BH\sigma$, where $\...
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Feynman's corrective term in his approach to the Onsager Problem

I am studying Feynman's book: 'Statistical Mechanics: A set of lectures' and his approach to the Onsager problem(Section 5.4). In the subsection 'Method of Calculating partition Function', Feynman ...
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Find partition function for a classical harmonic oscillator with time harmonic forcing

I have been trying to find partition function for classical harmonic oscillator with time harmonic forcing term and reached an expression. I want to know if I am correct. There is abundant ...
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181 views

What is the physical meaning of $W[J]=\frac{\hbar}{i}\ln Z[J]$?

The quantity $Z[J]$ (which is the generating functional for all Green functions) physically represents the probability amplitude for a system to remain in the vacuum state. Can we find a similar ...
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31 views

Finding the thermal expectation value of a magnetic system from the partition function

Consider two coupled magnetic systems. The Hamiltonian of this system is: $H_{eff} = \begin{bmatrix} H_{m_1} & U \\ U' & H_{m_2} \end{bmatrix}$. Each block is a $2\times 2$ Hamiltonian itself....
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64 views

Mean field solution of the Ising Model

I try to compute the variational free energy in the Ising Model using the bogoliubov inequality: \begin{equation} \mathcal{F}(\lambda) = F_{0}(\lambda) \ + \ \langle\mathcal{H_{1}(\lambda)} \rangle_{...
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54 views

Average value using partition function

Let's say I have 4 particles with energy levels $0\,\rm{eV}$ , $1\,\rm{eV}$,and two particles with $3\,\rm{eV}$ energy levels. If I want to find the average value of energy I can say that $$\bar{E}=\...
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Query about derivation for the grand partition function for an ideal fermi gas

I am having trouble understanding the following derivation: I am having a bit of trouble understanding what it means to sum over all states. So this is how I am interpreting the above. We have a ...
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While applying the principles of statistical mechanics to a photon gas, why do we use the partition function for a simple harmonic oscillator?

I'm trying to follow the derivation of the Stefan-Boltzmann constant in the Thermal Physics textbook written by Blundell and Blundell. And after deriving the density of states $g(\omega) d\omega$, the ...
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114 views

Path integral zero dimensional QFT

We consider the following partition function$$ \mathcal{Z}[\lambda] = \int{dx \; \exp\left(-\frac{1}{2}x^2-\frac{\lambda}{4!}x^4\right)} $$ Which is basically $\phi^4$ theory in 0+0 dimensions. The ...
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Why is the relation $Z_N = (Z_1^N)/ N!$ applicable only when $Z_1$ is much larger than $N$?

If the N particles are distinguishable, then we can write the N-particle partition function ZN as $Z_N = (Z1^N).$ For indistinguishable particles: $ZN =(Z1^N)/N!$ This result has assumed that it is ...
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Density of states (particle in a box) confusion

The partition function for the particle in a box problem is $Z = V(\frac{2m \pi kT}{h^{2}})^{\frac{3}{2}}$ And the maxwell-boltzmann distribution therefore is $n_{j} = \frac{Nh^{3}}{V(2 \pi mkT)^{\...
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78 views

Partition function index i

I'm confused on how the index in the partition function represents the microstate In the derivation, we calculated the number of microstates $ \Omega = \frac{N!}{\Pi_{i} n_{i}!}$ and I think this ...
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Relation between a partition function and Hilbert spaces

I'm studying the Jones Polynomial paper by Witten, and a bulk of the analysis involves computing various partition functions. He says, in section 4, that calculating the partition function over a sub ...
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66 views

Partition function of 2 particles connected by a spring

Consider a system composed of two, point-like particles connected by a linear spring, enclosed in a box. For simplicity, consider the system to be one dimensional. The energy of such a system is $E(...
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42 views

Probability distribution for photon momentum in canonical ensemble

Suppose I have an ideal gas consisting of photons, each photon has an energy $\varepsilon = cp$ where $p = |\vec{p}| = \sqrt{p_x^2 + p_y^2 + p_z^2}$. I have calculated the single particle partition ...

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