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What is the physical meaning of a Partition Function in Statistical physics?

In many places in statistical physics we assume the partition function. To me the explanations after partition functions are most of the times clear but always wonder why a partition function and what ...
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0answers
36 views

Is there a local canonical ensemble partition function for a Bose-Einstein gas?

The grand canonical partition function for a Bose-Einstein gas is $$ Z_{\text{grand bos}} = \exp \left( \sum_{j=0}^{\infty} -\ln \left( 1-e^{\beta(\mu-\epsilon_j)} \right)g_j \right) $$ where $\beta$ ...
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2answers
44 views

Multiplicity vs Partition function

I'm a little confused between all the different notations for the multiplicity and partition function. They're not the same thing, are they? I know that entropy can be expressed as $ S = k \ln\Omega ...
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1answer
19 views

potential energy of a segment of an ideal chain under external force

I want to construct the partition function of an ideal chain under applied external force. Let's say we apply force to both ends (with opposite sign) acting in z-direction and a segment of the ideal ...
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1answer
84 views

canonical ensemble that is quantum mechanical and continuous?

I do not understand what the following statements from Wikipedia mean For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as $$ Z = ...
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1answer
32 views

Mono-atomic gas particles coupled by spring forces don't care how many particles are involved?

I calculated the partition function of $N$ classical atoms of identical mass $m$ who all experience a mutual spring forces with identical spring constant $k$. The Hamilton is \begin{align} H = ...
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2answers
70 views

Partition function of a 3D vibrating string

Is the partition function of a 3D vibrating string a sum of discrete energies, an integral of an energy continuum, or both? $$ Z_{\text{disc}} = \sum_{k=1}^{\infty}g_ke^{-\beta E_k} $$ or $$ ...
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1answer
72 views

A trace formula of two noncommutative operators

In many cases of quantum many-body problems, the Hamiltonian $H$ can always be divided into two parts, i.e. $H_0$ and $H'$. In this occasion, one can systemically calculate the partition function ...
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1answer
33 views

Partition function is simply temperature if possible sub system energy is continuous?

Partition function is $$Z=\sum_j\exp\left(-\frac{\epsilon_j}{kT}\right)$$ a sum over all possible energy levels $\epsilon_1,\epsilon_2, ..., \epsilon_M$. There must be a finite number of choices ...
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1answer
221 views

About the factorial N! in the partition function

After reading these posts: Why is the partition function divided by $(h^{3N} N!)$? , What is the resolution to Gibb's paradox?, and some of these: http://arxiv.org/abs/1012.4111 , ...
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1answer
39 views

Einstein model for thermal capacity of solids and indistinguishability of the oscillators

Albert Einstein's theory of thermal capacity of a solids makes the assumption that a crystal is made up from oscillators which of course oscillate, in all three directions. Thus, for N atoms of the ...
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1answer
148 views

Difference in partition function of classical and quantum Ideal gas

First, I have read this question:What is meant by the term "single particle state" There is an analysis going on in my book (Mandle F. Statistical Physics) that has brought me in a ...
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4answers
248 views

Why is the partition function divided by $(h^{3N} N!)$?

When computing partition functions for classical systems with $N$ particles with a given Hamiltonian $H$ I've seen some places writing it as $$Z = \dfrac{1}{h^{3N} N!}\int e^{-\beta H(p,q)}dpdq$$ ...
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1answer
34 views

What is the definition of 'relative population' in context of partition function?

In statistical mechanics, what is the definition (or mathematical definition) when authors refer to relative population in the case of a classical particle system?
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1answer
47 views

is it necessarily true that the partition function $Z$ (with degeneracies) $ =1$?

The partition function with degnerate energies is $$\text{Z}=\sum _ig_ie^{{-E_i}/{k_BT}}.$$ Because the partition function Z is defined as the normalisation constant, does Z always = 1?
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2answers
78 views

Conceptual explanation of the Single particle partition function

The Single particle partition function is defined mathematically as $$\text{Z=$\sum $}g_ie^{\left(\frac{-E_i}{K_BT}\right)}$$ But what is the physical interpretation of the partition function and ...
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0answers
41 views

Computation of the partition function of a fermionic oscillator

I don't understand the following steps in the calculation of the partition function for a fermionic oscillator (Nakahara). The eigenvalues of the operator ...
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0answers
35 views

Fermionic oscillator and Hurwitz zeta function

Good resources for calculating Partition function of the fermionic oscillator using the Hurwitz zeta function? I liked the way Nakahara explain this, but some parts are really tricky for me (e.g. the ...
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1answer
45 views

Superstring vacuum amplitude on the torus

My question is how to obtain the superstring (Type II A and B) vacuum amplitudes on a torus. They are given in Polchinski's String Theory Vol. 2 equation (10.7.9): ...
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0answers
87 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
47 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
56 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
57 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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1answer
56 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
78 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
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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. ...
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1answer
42 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
28 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
59 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 ...
2
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1answer
146 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
39 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 ...
0
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1answer
65 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
168 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 ...
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1answer
150 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
52 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
56 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
92 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
35 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
198 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
717 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
46 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 ...
2
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1answer
213 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
250 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? ...
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1answer
298 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
163 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
114 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 ...
3
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0answers
61 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
106 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
106 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 ...
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2answers
258 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 ...