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8
votes
2answers
275 views

Why are the Nambu-Goto action and Polyakov action equivalent at quantum level?

It's a well known elementary fact that the Nambu-Goto action $$S_{NG} = T \int d \tau d \sigma \sqrt{ (\partial_{\tau} X^{\mu})^2 (\partial_{\sigma} X^{\mu})^2 - (\partial_{\sigma} X^{\mu} \partial_{\...
3
votes
1answer
58 views

Srednicki's Path Integrals: Ground-State to Ground-State Transition Amplitude in the Presence of a Perturbation

Srednicki's Quantum Field Theory mentions the following at the end of the unit on path integrals in non-relativistic quantum mechanics: Assume that the total Hamiltonian is of the form, $$ H = ...
25
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}\...
3
votes
1answer
647 views

Density of classical states in quantum theory

Let's first treat electrons as classical objects. I can evaluate the classical energy of each state in a configurational space (3N real numbers and, say, spins) using just Coulomb's law. Then I ...
0
votes
2answers
55 views

Canonical partition function and counting

That's a silly silly question, so my apologies, but in this moment I could not reach out! Let's have a system made of a particle reservoir $R$, and a subsystem $S$. The total particle number is $N$. ...
1
vote
0answers
40 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 ...
3
votes
2answers
47 views

Change of variables in canonical probability density

In K. Huang's book Statistical Mechanics, par. 7.2, the author writes the canonical partition function in a different way: $$Q = \frac{1}{N! h^{3N}} \int dp dq \ e^{-\beta H(p,q)} = \int_0^{\infty} ...
1
vote
0answers
54 views

Classical Grand canonical partition function derivation

Consider a classical grand canonical ensemble. Let $S_r$ be the reservoir entropy. Suppose it could be expanded at first order: $$S_r \approx S_r(E_t,N_t) + \frac{\mathrm dS_r}{\mathrm dE_i} \cdot ...
4
votes
2answers
105 views

Partition function and coherent state path integral

I have been working through the derivation of the partition function expressed as a path integral in terms of coherent states, following the many-body condensed-matter field theory books of Altland &...
1
vote
1answer
55 views

Partition sum for $SO(N)$ one-dimensional lattice model

I'm looking for derivation of explicit form of partition function for $SO(N)$ one-dimensional lattice model. The initial expression is $$ Z = \int \limits_{-\infty}^{\infty}d\sigma_{1}...d\sigma_{N}\...
3
votes
1answer
61 views

Physical interpretations of the generating functions $Z[J]$ and $W[J]$ (or $E[J]$)

In quantum field theory, the generator of all Green's functions $Z[J]$ and that of the connected Green's functions $E[J]$ are related as $$Z[J]=\exp[-iE[J]]=\int D\phi\exp[i\int d^4x(\mathcal{L}(\phi)+...
1
vote
1answer
40 views

What is the difference between these two expressions for the partition function, Z?

What is the difference between these two expressions given for the partition function, Z? $$Z = \sum_{i}e^{-\varepsilon_i/kT}$$ $$Z = \sum_{j} g_je^{-\varepsilon_j/kT}$$ where each energy level has ...
0
votes
1answer
58 views

Relation between Canonical and Grand Canonical Ensemble for a two levels system

For a two-level system of classical non interacting distinguishable paricles, labelling energy levels as $+\epsilon, -\epsilon$, in the canonical ensemble I found for the partition function Q$=2\cosh(\...
5
votes
2answers
345 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
53 views

Why does this formula for the partition function not include the multiplicity?

I am having problems understanding the formulas used for describing the partition functions and the probability distributions for canonical ensembles. In the first case I have two formulas for the ...
1
vote
1answer
49 views

How to write any partition function?

So I am familiar with the derivation of the partition function for a canonical and a grand canonical ensemble. I have seen definitions of the partition function for some of the quantum counterparts of ...
0
votes
0answers
29 views

3-state Potts model - probability of finding a site in state 1

Question: Consider the 1d 3-state Potts model of N sites (i.e., each site can be in either state 1, 2 or 3). Find the partition function and the probability of finding a site in state 1, $< \frac{1}...
1
vote
1answer
88 views

Deriving Enthalpy from Stat Mech

One can derive all the numerous thermodynamic potentials (Helmholtz, Gibbs, Grand, Enthalpy) by Legendre transformations, but I'm interested in seeing each from Stat Mech instead (ie taking the log of ...
1
vote
1answer
59 views

Difference between semi-classical Maxwell Boltzmann Statistics and Boson Statistics

Since semi-classical MB assumes the indistinguishability of particles and Boson Einstein statistics similarly treats degenerate states as indistinguishable states. What is their difference when ...
6
votes
1answer
140 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 ...
0
votes
1answer
38 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 ...
1
vote
1answer
35 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
vote
1answer
65 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 ...
5
votes
1answer
233 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 ...
1
vote
0answers
59 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 ...
3
votes
1answer
381 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....
0
votes
1answer
103 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: ...
2
votes
1answer
179 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 $[k,k+\...
2
votes
0answers
78 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' = ...
2
votes
1answer
121 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 D(\epsilon)d\...
3
votes
1answer
77 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 ...
38
votes
4answers
2k 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 ...
1
vote
0answers
27 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.
1
vote
0answers
470 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 ...
1
vote
0answers
62 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 $\beta$ ...
9
votes
4answers
557 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 $$Z = \dfrac{1}{h^{3N} N!}\int e^{-\beta H(p,q)}dpdq$$ ...
2
votes
2answers
251 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
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 ...
1
vote
1answer
109 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
votes
1answer
167 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
votes
1answer
39 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 = \dfrac{...
1
vote
2answers
110 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 $$ Z_{\text{...
1
vote
1answer
83 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 ...
0
votes
1answer
81 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 ...
2
votes
1answer
84 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 ...
1
vote
1answer
275 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 ...
13
votes
3answers
1k views

Is there a way to obtain the classical partition function from the quantum partition function in the limit $h \rightarrow 0$?

One would like to motivate the classical partition function in the following way: in the limit that the spacing between the energies (generally on the order of $h$) becomes small relative to the ...
1
vote
1answer
186 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
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?
1
vote
2answers
157 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 it'...