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1answer
31 views

Question on the degeneracies of a thermodynamic system

I think I have a basic understanding of what degeneracies are: Two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon ...
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1answer
31 views

Partion function for ideal gas - why use only one octant?

In these lecture notes (page 2) and in other sources I have checked, it says that the number of states with $k\in[k,k+dk]$ is: $$dN=\frac{4\pi k^2V}{8\pi^3}$$ Saying the factor of $8$ comes from the ...
1
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1answer
46 views

Relation between the $N$ particle partition function and probability?

For the 1 particle partition function the probability that the particle is in the state with energy $\varepsilon_i$ is given by: $$P_i =\frac{e^{-\varepsilon_i \beta}}{Z_1}$$ where $Z_2$ is the 1 ...
4
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1answer
128 views

General formulation for fermions

Let us look at a set of fermionic creation and annihilation operators $b_n$, $b_n^\dagger$ with $n \in \mathbb{N}$. What is the precise relationship between this and this sequence here (partition ...
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0answers
35 views

What is the phase space volume in terms of angular momentum?

Given a rigid rotor Hamiltonian, defined along the principle axes as $$ H = \sum_{i=1}^3 \frac{L_i^2}{2I_i} $$ say we would like to compute the classical partition function of this system. Is the ...
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1answer
56 views

Statistical mechanics of ideal gas in a box with adsorption states on surface [closed]

Assume we have a cubic box of side length 1m with ideal gas particles inside. We assume the binding energy of a gas molecule to the wall is 1eV. One can make the simplifying assumption that: ...
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0answers
74 views

How to find explicit QFT particles from a heterotic String theorie partition function

Starting with a SUSY $E_8$ x $E_8$ heterotic partition function: $ Z = - \frac{1}{8} \sum Z^8_x Z_4[^s_{s'}] Z_8[^t_{t'}] Z_8[^u_{u'}] $ where the sum is over all spin structures $s,t,u,s',t',u' = ...
5
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1answer
189 views

What are alternative ways to think about transfer matrix as used in Ising model?

I recently learned about how to find the partition function of Ising model using Transfer Matrix method. At my level of understanding things, it is a trick that happens to work! I would like to ...
3
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1answer
65 views

Does the canonical partition function count microstates?

The microcanonical partition function is the density of states. The canonical one, from a dimensional point of view, is still a number of states, but does it actually count microstates? I tried ...
2
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1answer
87 views

Partition Function and BlackBody Radiation

I'll start with a few definitions: $$\beta \equiv \frac{1}{k_bT}$$ Where T is the temperature of a system. And the partition function: $$Z \equiv \sum_{j}e^{-\beta \epsilon_j}=\int ...
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0answers
25 views

Partition Function And Macroscopic Properties

In renormalization group transformations, partition function is fixed. My question is which thermodynamic properties are fixed in a renormalization group transformation.
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0answers
252 views

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
51 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 $\beta$ ...
2
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2answers
110 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 ...
0
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1answer
30 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 ...
1
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1answer
98 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 = ...
0
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1answer
38 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
90 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 $$ ...
1
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1answer
80 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
65 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 ...
3
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1answer
322 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 , ...
2
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1answer
68 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
222 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
376 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
109 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?
0
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1answer
58 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
112 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
50 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 ...
0
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0answers
39 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
53 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
139 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 $g_i$ ...
2
votes
1answer
138 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
66 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
60 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
0
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1answer
112 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 ...
0
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1answer
93 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. ...
0
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1answer
54 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
35 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
62 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
votes
1answer
184 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: $$\frac{1}{N}\ln Z ~=~ \ln(2 \cosh^2(\beta J)) $$$$+ ...
2
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0answers
50 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
89 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
226 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
161 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
63 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
65 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 ...
3
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1answer
138 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 ...
0
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0answers
41 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 ...
0
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0answers
392 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 ...