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9
votes
4answers
544 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 $$Z = \dfrac{1}{h^{3N} N!}\int e^{-\beta H(p,q)}dpdq$$ ...
24
votes
4answers
2k 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. $$p_j=\frac{1}{Z}\...
5
votes
2answers
337 views

Physical meaning of partition function in QFT

When we have the generating functional $Z$ for a scalar field \begin{equation} Z(J,J^{\dagger}) = \int{D\phi^{\dagger}D\phi \; \exp\Big[{\int L+\phi^{\dagger}J(x)+J^{\dagger}(x)}\phi\Big]}, \end{...
2
votes
1answer
210 views

Quantum Stat-Mech Proof of an Inequality for the Partition Function

I have the following problem that I was unable to solve for class, but I had a couple first steps that I started with that I am unable to finish. I know I can't get this since it's already been ...
4
votes
1answer
4k 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 can'...
10
votes
3answers
1k 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 $$ Z(\beta) = \int e^{-\beta H(x)}\,\mathrm{d}x $$...
3
votes
1answer
375 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 , http://bayes....
4
votes
3answers
458 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 ...
3
votes
2answers
1k views

Canonical partition of a boson gas

I have a 1D gas made of $N$ particles placed in a harmonic potential well, so the Hamiltonian is: $$ \mathcal H = \sum_{j=1}^N \left ( \frac{p_j^2}{2m} + \frac{1}{2}m\omega^2 x_j^2 \right )$$ The ...
4
votes
1answer
122 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 ...
4
votes
2answers
490 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 ...
1
vote
1answer
185 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.
1
vote
0answers
39 views

Link Between the Density Operator and the Partition Function and Boltzmann Distribution in Quantum Statistical Mechanics

I have a very limited knowledge of statistical mechanics, but I seem to running into some related concepts for my background readings for the research project this summer. For example, see the ...
0
votes
1answer
40 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 ...
0
votes
1answer
62 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?