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Questions tagged [statistical-mechanics]

The study of large, complicated systems by means of statistics and probability theory, in order to extract average properties and to provide a connection between mechanics and thermodynamics.

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Sum to an integral in deriving equipartition theorem

I'm reading this derivation of the equipartition theorem for ideal gases. On the second page, it is mentioned that the partition function as a simple sum, $${\displaystyle Z=\sum _{i}e^{-\varepsilon ...
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61 views

Finding the correlation function using the expression of the free energy (Ising model, Landau theory)

I am working on a homework problem regarding the Lee Yang theorem, though my issue already exist using only the standard approach to the Ising model. Simply put, i have no idea how to explicitly ...
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1answer
52 views

Internal energy of ideal gas from statistical mechanics

I'm following this derivation of the equipartition theorem: http://vallance.chem.ox.ac.uk/pdfs/Equipartition.pdf On the second page, it is said that a standard result from statistical mechanics is ...
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1answer
33 views

Boltzmann equation derivation for $H=v\sigma \cdot p$ hamiltonian

I am trying to write the Boltzmann equation for $$H=v_{F}\vec{\sigma}\cdot(\vec{p}-e\vec{A}).$$ This is a free charged particles gas. The velocity for this hamiltonian is $$\vec{v}=v_{F} \vec{\sigma}.$...
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Fluctuation-dissipation theorem for velocities

I am given the following problem about fluctuation dissipation theorem: Consider an external force $f(t)= \frac{f_0}{2}(e^{i\omega_0 t}+e^{-i\omega_0 t})$ acting on a particle with momentum $p=mv$ in ...
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1answer
49 views

Understanding what the Bose-Einstein distribution

I'm currently studying Kittel's Solid State Physics and in his chapter on Phonon heat capacity, we need to first calculate the total energy $U$. Phonons have energy $E_n = (n+1/2)\hbar\omega$ and he ...
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1answer
57 views

How to calculate the time average of the mean square displacement?

I’m trying to understand the mean square displacement (MSD) as calculated in time-averaged single-particle tracking experiments. For simplicity, I’ll consider the 1D case. Following this article, the ...
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1answer
148 views

The entropy of lottery drawing machines

Suppose we have a fair lottery drawing machine where you have a container of numbered balls that is rotated many times such that interaction of the balls with themselves and the container produces a ...
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64 views

What is finite size scaling theory?

The question is in the title. I read some articles about it but I could not understand it completely. What does it mean when we say scaling of any system? Can you please give a brief introduction ...
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65 views

How does renormalization relate to emergence?

In statistical mechanics renormalization is often related to coarse-graining which in turn allows to calculate some macroscopic states. The resulting macroscopic description is sometimes called ...
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1answer
73 views

Degeneracy of Maxwell-Boltzmann distribution

This question was already asked here: Degeneracy in Maxwell Boltzmann distribution But the answer was not very satisfying so I'm asking again. I can somewhat understand the derivation of Maxwell-...
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1answer
78 views

Path integral formulation of the density matrix ρ

In Feynman's Statistical Mechanics - A Set of Lectures, upon the introduction of the path integral, a series of approximations are made in order to calculate integrals. I am unsure how exactly to get ...
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Why we can not observe phase transition in finite systems? [duplicate]

In condense matter physics books it is mentioned that we can not observe phase transition in finite systems. Why is it so?
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Physical Significance of $U$ (Internal Energy ) , $H$ (Enthalpy) , $F$ (Free Energy) and $G$ (Gibbs Free Energy)? [closed]

I know their mathematical definitions and how these terms are interrelated (mathematically) but I fail to understand the physical meaning of none but one which is INTERNAL ENERGY . It seems ...
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1answer
38 views

Why can't we partition the quantum-mechanical phase space into discrete cells?

Near the bottom of page 2 in this paper, von Neumann states that not only is it impossible to simultaneously measure $x$ and $p$, but also that it is impossible to partition the quantum-mechanical ...
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51 views

Phase space in Statistical mechanics

I'm currently struggling with the concept of phase space in Thermodynamics/Statistical physics. In particular I have trouble understanding the use of the "one-particle phase space". If we look at a ...
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1answer
77 views

Finite temperature QFT: Can vertex correction function/ initial correlations induce short-time tunneling?

The collision term in the Kadanoff-Baym equation has the structure $I(\tau_1,\tau_2) = \int_C d \tau'\Sigma(\tau_1,\tau') G(\tau',\tau_2)$ where the contour $C$ is along the time-Forward $C_+$, time-...
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1answer
40 views

Fluctuation of number of particles in one state in canonical ensemble

I haven't touched statistical physics for a while and am stuck in quite a basic question, and surprisingly, I cannot find any information in the internet that helps me to think it through. What is ...
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1answer
42 views

When exactly do identical fermions interact?

For the case of $N$ identical fermions in a three-dimensional box, the Pauli Exclusion Principle necessitates that the overall wavefunction of the system is antisymmetric. No two fermions can occupy ...
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Computing a 'polygonal loop' diagram

I consider a simple spherical SK model of the type $H = -\frac{1}{2} \sum_{i,j} J_{ij} X_i X_j$, with $J$ symmetric, of sieze $n$. Suppose I have $p \in \mathbb{N}^*$ (assume $p \ll n$). I want to ...
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1answer
50 views

Why do we need large system in Monte Carlo simulations?

Why do we need to simulate large systems (especially in spin systems)? Is it just a better representation of the real behaviur of magnetic systems? Is there an instability of the finite system with ...
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1answer
92 views

Why Nearest level spacing distribtuion $p(s)$ in RMT is so much popular?

In Random Matrix Theory, Question 1): Two Distributions are very popular. 1) The density of Energy levels $\rho(E)$. 2) Distribution of nearest energy levels spacing $p(s)$. I understand why ...
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$u=(3/2)nRT$ when $T=0$

Why does a body with $0\ K$ temperate does the internal energy NOT become $0\ J$? My lecturer said it had to do with some quantum mechanics, but I never really got any answers from him. I really want ...
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Weird and unintuitive results from thermodynamics/hydrostatics for an isothermal atmosphere and possible explanation(?)

It is known from hydrostatics that for a fluid in equilibrium in a gravitational field, $$\frac{dP}{dz} = -ρg$$ Let us from now on suppose the atmosphere is isothermal and has temperature $T$. We ...
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1answer
35 views

Irreversibility and increase of number of microstates

We know,in an irreversible process,always entropy of an isolated system increases,from Clausius inequality.Again,entropy is the logarithm of the number of accessible microstates of a system.How can we ...
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Quantum micro canonical ensemble

In Huang's Statistical Mechanics, the quantum micro canonical ensemble is introduced in an unorthodox way. Here, the isolated system of the classical ensemble is supplemented by an external reservoir (...
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1answer
64 views

Frustrated Ising model

Consider a 2D Ising model with nearest neighbour, and second nearest neighbour interactions $\mathcal{H}= -J_1\sum_{\langle ij\rangle}\sigma_i \sigma_j-J_2\sum_{\langle\langle ik\rangle\rangle}\...
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50 views

Is it necessary to assume that equivalent microstates cannot get transformed into inequivalent microstates to derive the Landauer principle?

The Landauer erasure principle states that to erase a bit of information from a system, the entropy of the environment will be increased by at least $k_B\log2$, or equivalently, it costs at least an ...
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Difficulty in calculating effusion rate using the kinetic theory of gases and statistical mechanics

I have tried multiple sources and methods, but my attempts at a proof of the number of particles leaving a gas using statistical mechanics keep finding the same wrong result. I have tried to read ...
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48 views

Broken symmetry Ising model

Consider an Ising model for a 2D square lattice where we have interactions for, let's say nearest neighbours and second nearest neighbours $$\mathcal{H}=-J_1\sum_{\langle ij\rangle}\sigma_i\sigma_j-...
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1answer
50 views

What is the difference between non-equilibrium and equilibrium phase transitions?

My question is about the distinction between certain kinds of phase transitions. I understand what the difference between first and second order ones are. What is the difference between non ...
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1answer
98 views

Mean field 3d Ising model in Landau theory

I am solving a problem for some course I am following. I am given the Hamiltonian: $$H = 3NJ \langle \sigma\rangle ^2 -6J\langle \sigma\rangle \sum_i\sigma_i,$$ where $i$ sums over all sites ($=N$)....
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Why does increasing the volume in which a gas can move increase its entropy?

Let's say we have a box with a non-permeable wall separating the box in half. There is gas on the other side of the wall. Now we remove the wall so that the gas can diffuse to the other half of the ...
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Why repulsion between adjacent levels disappear in Poisson distribution?

In Poisson distribution $P(s)= exp(-s)$ why when spacing s between two energy level is zero P(s) is 1 means there is maximum probability that two levels are close to each other. Can I say that ...
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Can we observe the cross over between different Dyson ensemble distributions at the same time?

In random matrix theory we randomly choose matrix elements But they should follow certain symmetry properties. According to Wigner there can be 3 distributions GOE,GUE,GSE and these distributions are ...
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Entropy and equilibrium concepts at astronomic scales

I am always puzzled to read here and there discussions dealing with thermodynamic concepts applied to astronomic scales where gravity matters. To my opinion, there is a certain carelessness to go into ...
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What is the link between superfluids and BEC?

i’m studying superfluids (in particular $^4 He$) and one of the first theorical apporoach was with Bose-Einstein condensation and i know that we can calculate the $T_c$ and it is close to the the ...
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2answers
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Can anyone explain equivalence of statistical entropy and thermodynamic entropy?

I read on wikipedia how Clausius came to define entropy after studying the Carnot cycle (He found a relation between heat transfer and temperature which was a state function,and named it entropy) but ...
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1answer
66 views

Can indistinguishable microstates evolve into different macroscopic states?

I guess this question is more about definition than about any physical principle. Can a given physical system (or can there be a physical system) admit microstates that have identical macroscopic ...
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69 views

How to quantify the mixing of two hard-spheres gases?

Suppose that there are two types of hard-spheres gases ($a$ and $b$) in a box. Suppose that their radius is much smaller than the box's characteristic size. These two gases tend to repel each other. ...
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1answer
78 views

Why are simulations like Monte Carlo or Metropolis studied for Ising Models when 1d and 2d case have analytical solutions?

I know that absolute analytical solutions exist for the 1d and 2d case but need some intuition as to why these simulation algorithms are used and how do we benefit from them ?
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1answer
47 views

Specifying the initial nonequilibrium distribution $f(\textbf{r},\textbf{v},t)$ in Boltzmann equation?

Within the single relaxation time approximation, the collision term in the Boltzmann equation is approximated as $$\Big(\frac{\partial f}{\partial t}\Big)_{\rm coll}=-\frac{(f-f_{\rm eq})}{\tau}$$ ...
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1answer
38 views

Why does the Boltzmann equation deal with single-particle phase space density?

Why does the Boltzmann equation deal with single-particle phase space density $\rho_{1}(\textbf{r}_1,\textbf{p}_1,t)$ rather than the N-particle phase space density $\rho(\{\textbf{r}_i,\textbf{p}_i,t\...
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1answer
40 views

I reasoned that if $T_2 > T_1$ then $E_1>E_2$. Obviously this can't be right, what is the flaw in my logic?

We have an entropy function and I've shown that for this function $T_1(E_1)=T_2(E_2)$. Further, the fact that the entropy function is concave is stated. There are two systems in contact and only ...
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2answers
85 views

Is uncertainty and correlations actualy the same thing?

In this paper on page 2 it is said that The entropy $S(\rho_A)$ measures the amount of correlation (classical and/or quantum) between $A$ with the external world. Now this is confusing me a little ...
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1answer
53 views

Why is the number of excited vibrational modes $g(\nu)d\nu$ proportional to $x^2e^{-x}$ in Debye's theory?

I come across a problem in Terrell Hill's "Introduction to statistical thermodynamics" saying that: In the Debye theory, the number of excited vibrational modes in the frequency range $\nu$ to $\nu+...
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1answer
61 views

Derivation of Proportionality of Phase Space Volume log(Γ)∝N

In the derivation of extensivity of entropy for the micro-canonical ensemble, we assume an ensemble of two systems with the energies $E_1$ and $E_2$. The total energy is given as $E<E_1+E_2<E+\...
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1answer
65 views

How can I derive the analog of the susceptibility sum rule for the specific heat?

How can I derive the analog of the susceptibility sum rule for the specific heat? Does an infinite correlation length imply an infinite specific heat? $$ \chi = \frac{\partial M}{\partial H} = \frac{1}...
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Density Fluctuation in N-Particle Brownian Motions

I am studying spatial population movement and would like to model the density fluctuation by assuming a Brownian movement for each individual. Because the total number of individual ($N$) is large but ...
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13 views

$- \mu / kT $ equal to $1/N$ in Bose Einstein Condesation

In what conditions $ -\mu/ kT = 1/N $ so that we can write $ e^{\mu n /kT} = e^{-n/N} $ in Bose Einstein condensation