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41 views

How is partition function related to ordinary generating function?

Ordinary generating function can be used to solve combinatorial enumeration problems. Now if the energy levels are discrete, say $g_i$, and if one want to count how many ways one can add up $g_i$ ...
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1answer
17 views

How to understand Density of States with dispersion relation

I am having trouble understanding the Density of states concept. As I currently understand it, for the density of states $g(k)$ it is the number of microstates with wave number in the range ...
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0answers
39 views

Proving the Virial theorem

Consider the expectation in the canonical ensemble defined by $$\left\langle x_i\frac{\partial \mathcal{H}}{\partial x_j} \right\rangle=\frac{1}{Z}\int d\Gamma x_i\frac{\partial ...
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1answer
51 views

Where does this hyperbolic tangent in Nakahara's text come from?

I don't see why the term with $\tanh$ appears in the equation 1.164 The textbook is the second edition of Geometry, Topology and Physics
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0answers
25 views

How do I find the partition functions for the Saha equation for elements that are complex like Argon?

The question I have is with regards to the partition function needed for solving the Saha equation for Argon. Basically the part I'm confused with is the part of the formula where $g_i/g_a$, where ...
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1answer
65 views

Relation between 1D and 3D partition function

In a solid state physics text author said $$Z_{3D} = [ Z_{1D} ]^{3} $$ $$<E_{3D}>=3<E_{1D}>$$ Where $Z$ is partition function. Can anyone convince me why it is so?
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4answers
1k views

The unreasonable effectiveness of the partition function

In a first course on statistical mechanics the partition function is normally introduced as the normalisation for the probability of a particle being in a particular energy level. ...
0
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1answer
34 views

Internal energy from a canonical partition function

How to derive this relation, though it looks simple in eyes $$U=-\frac{\partial ln Q}{\partial \beta}$$ where Q is a canonical partition function and $\beta=\frac{1}{K_BT}$
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0answers
22 views

Given the Pair Correlation Functions for a Multi-component Fluid, is There a Simple Way to Obtain the Helmholtz Free Energy?

I think it may not be possible. Given all the pair correlation functions (e.g. calculated from Ornstein Zernike theory), it seems possible to obtain the internal energy (assuming different species ...
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0answers
49 views

Ising Model with All Spins Interacting with All Other Spins

I am studying the Ising model with all spins interacting with all other spins and have formulated $Z$. I am trying to understand what it means to study at large N but not infinite N. I know that at ...
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1answer
109 views

Expansion of Onsager's Exact Partition Function for 2D Ising Model

We have a question where we are given the exact expression for the 2D Ising model partition function: $$\frac{1}{N}\ln Z ~=~ \ln(2 \cosh^2(\beta J)) $$$$+ ...
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0answers
29 views

manipulations with SU(N) Nekrasov partition function

Think of a Young tableau $R$ as collection of rows $y_1 \geq ... \geq y_d > y_{d+1}=0$ and all others zero, with $\ell(Y):= \sum_j y_j$ and for a box $s=(i,j)\in R$ we have $a_Y(s):=y_i-j$ and ...
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1answer
52 views

Heat capacity $C$ at low temperature

The internal energy, $U = Nk_bT$ where $N$ is particle number, $k_b$ is Boltzmann constant and $T$ is temperature. Therefore, the heat capacity $C$ is given by $C=\frac{dU}{dT}=k_b$. However, in ...
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1answer
129 views

Why is the canonical partition function an exponential?

It makes intuitive sense that micro-states of higher energy occur with a lower probability and the exponential function has reasonable properties. However can a physical explanation be given to why ...
6
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1answer
131 views

Uses for Action from Lagrangian Mechanics

In my course on Lagrangian/Hamiltonian mechanics I noticed that we dealt with finding the stationary point of the change in action $ \delta S $ and we were never really doing anything with $ S $ ...
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0answers
46 views

Is there a well-defined partition function of 4d Yang-Mills?

So I've looked everywhere to find a resource on 4 dimensional Yang-Mills partition functions, but have only managed to find examples using supersymmetry. Is there a resource describing the partition ...
1
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1answer
50 views

Canonical Distribution (Partition Function)

For the canonical distribution $$ w_{n}=e^{(F-E_{n})/T}, $$ is the sum $$ Z=\sum_{n}e^{E_{n}/T} $$ a sum over energies or a sum over states? Perhaps this is a silly question, but Landau and Lifshitz ...
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1answer
61 views

Grand Canonical Partition Function

I'm looking over posted lecture notes for a course, and this derivation of the Grand Canonical Partition function eludes me. It goes like this: occupation numbers $n_{α}=0,1,…$, Total particle number ...
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0answers
32 views

How can one approximate integral def. of Z by the max value of the integrand?

I am taking a course in statistical physics, and while reviewing my notes from the lectures I came across something that I cannot get my head around. We arrive at an integral expression for the ...
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0answers
118 views

Partition function microcanonical ensemble

I was wondering if there is a way to understand the partition function for a microcanonical ensemble $$\mathcal Z(E)=\sum_{\text{microstate $i$ with energy $E$}} w_i$$ as a limit of the continuous ...
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3answers
480 views

Why is the canonical partition function the Laplace transform of the microcanonical partition function?

This web page says that the microcanonical partition function $$ \Omega(E) = \int \delta(H(x)-E) \,\mathrm{d}x $$ and the canonical partition function $$ Z(\beta) = \int e^{-\beta H(x)}\,\mathrm{d}x ...
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0answers
39 views

CFT calculation of the Partitionfunction of $2+1$ dimensional gravity

I want to reproduce formula (4.29) in http://arxiv.org/abs/0804.1773v1 given by: $$ Z=Tr(q^{L_{0}}\bar q^{\bar L_{0}})=|q|^{-2k} \prod^{\infty}_{n=2}\frac{1}{|1-q^{n}|^{2}} $$ Where the trace is ...
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1answer
147 views

Expectation value of $a_i^\dagger a_i$ for thermal density matrix

Suppose we have some heat bath with Hamiltonian, $$H=\sum_n \left(a^{\dagger}_na_n+\frac{1}{2}\right)\hbar\omega_n$$ and a density matrix $\rho=Z\exp(-\beta H)$ for some normalisation $Z$. ...
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1answer
179 views

Calculation of the partition function for a classical 2D gas lying on the surface of a sphere of constant radius $R$

I'm kind of confused with this system. My first question is. Is the Hamiltonian of one particle of this gas the following? ...
4
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1answer
240 views

Strange definition of microcanonical partition function

I always thought that the microcanonical partition function would measure the number of states that correspond to some fixed energy. Despite, I found in this paper (equation 3.4) that we integrate ...
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3answers
136 views

Proof of Connected Diagrams

If $Z[J]$ is the generating functional for the path-integral, could any prove (or more reasonably, refer me to a proof) that $$W[J]\equiv\frac{\hbar}{i}\log\left(Z[J]\right)$$ "generates" only ...
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0answers
81 views

Fluctuation spectrum of lipid bilayer membranes

I am interestend in calculating the fluctuation spectrum of a thermally fluctuating 2d membane which is only subject to a surface tension $\sigma$. ($\mathcal{H}=\sigma\int\mathrm{d}A$) Depending in ...
2
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0answers
46 views

Is there a reasonable lower bound for free energy per site of the 2D Ising model in the presence of an external field?

Given the standard Ising partition function: $$Z(\theta ,h) = \sum\limits_{\bf{x}} {\exp \left\{ {\theta \sum\limits_{(i,j) \in E} {{x_i}{x_j}} + h\sum\limits_{i \in V} {{x_i}} } \right\}}, $$ is ...
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0answers
99 views

Wilson's Renormalization Group and Lie's Third Theorem

If you think of a one-parameter group of transformations along a curve in the plane as a (Lie) group, and the tangent vector to the curve as a generator of the curve we can intuitively understand ...
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1answer
90 views

Partition function for composite systems

I want to understand the derivation of the partition function for two distinguishable non-interacting particles. Let the energy of particles $1$ and $2$ be $E_1$ and $E_2$ respectively. Setting ...
6
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2answers
200 views

Classical and Semi-classical treatments of the ideal gas

In the semi-classical treatment of the ideal gas, we write the partition function for the system as $$Z = \frac{Z(1)^N}{N!}$$ where $Z(1)$ is the single particle partition function and $N$ is the ...
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2answers
80 views

Partition Function - TOTAL internal energy vs Average Energy

Given a partition function Z, the books sometimes uses the terms 'total internal energy' and 'average energy' exchangeably. It confuses me to no end. On one hand they say that Internal energy is ...
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0answers
127 views

Understanding the product of partition functions by making sense of the maths and the physics

I have $N$ distinguishable particles in a 1D harmonic oscillator potential with 'proper' frequency $\omega$. The particles also have internal spin-$\frac12$ degrees of freedom in a magnetic field $B$ ...
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1answer
139 views

How to prove useful property of logarithm of generating functional in QFT?

How to prove that $\ln(Z(J))$ generates only connected Feynman diagrams? I can't find the proof of this statement, and have only met its demonstrations for case of 2- and 4-point.
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1answer
91 views

Same partition functions, different theories

I am reading the book "Basic Concepts of String Theory" by Blumenhagen, Lust and Theisen and in page 290 they say: "It follows that the $E8\times E8$ and the $SO(32)$ heterotic string theories have ...
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0answers
217 views

Free Energy of N Spin 3/2 Particles

This question is from the book "Introductory Statistical Mechanics" by Bowley and Sanchez. The question is as follows: Calculate the free energy of a system with N particles, each with spin 3/2 with ...
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3answers
331 views

Completing the square for Grassmann variables

When working with path integrals of both bosonic and fermionic field variables, I'm a bit unsure of how to do the usual complete the square trick when an interaction between the two is concerned. Say ...
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1answer
127 views

Partition functions in $\phi^{4}$ theory

The partition function in a $\phi^{4}$ theory is written \begin{equation}Z[J]=\int D\phi \, e^{-\int d^{4}x \left(\frac{1}{2}\left[(\nabla ...
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4answers
1k views

How exact is the analogy between statistical mechanics and quantum field theory?

Famously, the path integral of quantum field theory is related to the partition function of statistical mechanics via a Wick rotation and there is therefore a formal analogy between the two. I have a ...
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1answer
163 views

Separation of perturbative and non-perturbative contributions in partition function computation

The following is defined, where $\epsilon \to 0^+$ is a cutoff: $$ \mathcal{F}(Z)=\int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \frac{1}{\sinh^2 s/2} e^{-sx}. $$ Question: how do we see that ...
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0answers
64 views

What is known about the statistical mechanics of systems with normally distributed energies?

Consider a system taking on N states with energies $\epsilon \sim \mathcal{N}(\mu,\sigma^2)$. Are such systems well-studied in any context? I ask because I'd like to be able to take certain ...
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1answer
2k views

How do you determine the value of the degeneracy factor in the partition function?

In the partition function, expressed as $$Z = \sum_j g_je^{-\beta E_j}$$ I'm wondering what determines the $g_j$ factor. I've been trying to look around the internet for an explanation of it but I ...
0
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1answer
67 views

Can a distribution with sharper energy maximum than the exp-function give an equivalent theory?

Because for many particles the distribution $\varrho\sim\mathrm e^{-\beta\ H}$ has an extremely sharp maximum, the expectation values of the canonical ensemble agrees with that of the microcanonical ...
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3answers
773 views

The Chern-Simons/WZW correspondence

Can someone tell me a reference which proves this? - as to how does the bulk partition function of Chern-Simons' theory get completely determined by the WZW theory (its conformal blocks) on its ...
4
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2answers
307 views

Continuous phase transition only hold for infinite systems. Real systems are finite, hence, a paradox

Second-order or continuous transitions are usually identified with non-analyticies within the free energy (which is proportional to the logarithm of the sum of exponentials). Such singularities are ...
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0answers
120 views

How to understand the matrix behind a Hamiltonian?

thanks to the answers I received to my previous questions, I could derive correctly an elegant partition function for my problem which resembles a second quantized model taking the particles to be ...
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1answer
97 views

partition function for Wightman and Haag-Kastler QFT

From what I hear, some modern mathematical approach quantum field theory uses the following definition "A $d$-dimensional $S$-structured quantum field theory $Q$ is a mathematical object, ...
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0answers
47 views

Ascertaining a mathematical equality to derive a partition function

we have an equation like this: $$\mathcal N(x)=\sum_{q=1}^\infty (\psi(x,q) \log(q)) \qquad (1)$$ while $\psi(x)$ is the function for some oscillations (may contain complex part), $x\in \Bbb R$ and ...
2
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0answers
117 views

Number theoretical function applied in physics? [closed]

I have a series of number theoretic phenomena (mathematics) that I can describe exactly by the superpositions or linear combination of the below function (I know it is an inverse Fourier type). Does ...
2
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0answers
68 views

About deriving the multi-trace index in terms of the single-trace index

This question is in reference to this paper Combining their equations 5.2, 5.3, 5.6 and 5.7 one seems to be looking at the integral/partition function, $Z(x) = \prod_{n=1}^{n =\infty}\left [ \int ...