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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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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
40 views

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
46 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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1answer
42 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
50 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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Decimation, Ising chain, recursion relations

I'm having trouble understanding how to derive the recursion relations for the 1D Ising model. If we consider the Ising model for a 1D chain, we have the Hamiltonian $\mathcal{H}=-J\sum_{i}\sigma_i\...
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1answer
42 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
37 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
36 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 ...
2
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1answer
58 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
43 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
48 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
54 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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67 views

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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19 views

Can some one explain following formula for average impulse of a molecule in the derivation of Boltzmann–Gibbs distribution

Thus, the total average impulse contributed by a molecule with an $x$-component velocity ranging between $v_x$ and $v_x+\mathrm dv_x$, is given by $$2mv_x\cdot\frac{v_x\tau}{L_x}\cdot\sqrt{\frac{\...
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$- \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
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1answer
45 views

Escaping metallic solids under extreme pressure

Suppose you have metallic solids inside an indestructible tube, with a very powerful and indestructible piston - the piston gets a tiny hole on the piston. What would happen if you compress the solid ...
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0answers
15 views

why for two level system we consider both energy level while finding number of bosons in ground state?

Let suppose we have N number of particles in two level system. .Effective number of cobosons in ground state is $ <n_0>$ that can be written as $<\hat{n}_0>= Tr [\hat{n_0}\rho ]$ where $\...
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19 views

Partition function and probability in finite harmonic trap

Let suppose we have $N$ number of cobosons in 3D harmonic trap. The effective number of cobosons in some state m is $\langle n_m\rangle$ that can be written as $\langle\hat{n}_m\rangle= \mathrm{Tr} [\...
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Kosterlitz-Thouless transition and correlation function

I’m studying Kosterlitz transition on this book: https://tinymachines.weebly.com/uploads/5/1/8/8/51885267/kardar._statistical_physics_of_fields__2007_.pdf#page173 . At page 165 it says:” The gradient ...
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46 views

What is meant by finite harmonic oscillator?

What does it mean to take finite harmonic oscillator, In research article "http://iopscience.iop.org/article/10.1088/1367-2630/17/11/113015 ", we were finding effective number of cobosons in ...
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36 views

Can we say that density matrix gives probability?

In statistical physics Boltzmann probability is given by $$P= \frac{\exp(-\beta E_i)}{\sum_i\exp(-\beta E_i)}$$ whereas we can also write it $$\rho= \frac{\exp(-\beta H_0)}{\sum_i\exp(-\beta H_0)}.$$...
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47 views

Spin spin correlation function in topological phase transition?

during my vacation i have decided to study Kosterlitz and thouless phase transition (i have already posted 2-3 questions about that). I don't know quantum field so I did not expect to understand ...
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10 views

What is bose enhancement factor?

I am studying about composite bosons ,when we write N state for composite bosons in terms of its constituent particles, we add factor of normalization constant \chi in denominator.This normalization ...
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1answer
17 views

Integrating Carnahan-Starling Pressure

Given the Carnahan-Starling equation of state for a solution of hard-spheres, $$ Z = \frac{P}{\rho k_BT} = \frac{1 + \eta + \eta^2 - \eta^3}{(1-\eta)^3}$$ where $\rho = N/V$ is the number density and ...
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0answers
35 views

Laplace Transform Density of States & Partition function

I am currently going through Pathria's Statistical Mechanics text , and under the Canonical Ensemble description, the author stresses that the partition function of a continuous system is the Laplace ...
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1answer
31 views

Computing the average energy $\overline{E}$ in thermodynamic equilibrium for a paramagnet

I am having trouble with the following physics exercise: In the case of a paramagnet in a magnetic field $H$, show that from the requirement that $F(E)$ (the Helmholtz free energy) be minimal in ...
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0answers
65 views

Is square of probability $p^2$ is less than probability $p$? [migrated]

suppose we have $N$ possible states in system. There is a probability $p_n$ that system is in state $|n\rangle$, and the sum of all probabilities is one. Is there any general rule in math or physics ...
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Calculating the density of states for a free particle [closed]

I would like to compute the density of states $\rho(E)$ for a free particle with energy $\epsilon = h^2k^2/2m$, which exists in one and two-dimensions. This is in a statistical mechanics book, and I ...
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1answer
60 views

Canonical partition function for different systems

As a homework exercise for Advanced Statistical Mechanics I need to derive the canonical partition functions for the following systems: Single component ideal gas on a square lattice Single component ...
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1answer
72 views

What is the link between statistical and QFT correlation functions?

I'm studying statistical mechanics in particular correlation function: https://en.wikipedia.org/wiki/Correlation_function_(statistical_mechanics) and I have understood it. Now searching on internet ...
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42 views

Computing the Helmholtz free energy for a harmonic oscillator

I would like to calculate the helmholtz free energy for a harmonic oscillator, which has energies given by the equation $\epsilon_{n} = n\hbar\omega$, where $n$ is a natural number greater than or ...
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0answers
18 views

Venturi effect on molecular level

Is there a good explanation why the pressure becomes lower on the narrow part of the venturi pipe? I'm not interested in the newton explanation or pressure differences explanation. I want to ...
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2answers
48 views

In Boltzmann distribution, why is the system at the same temperature as the reservoir?

Consider a boltzmann distribution where the total energy of the reservoir and the system is $E$. The energy of the system can be $\epsilon_i$ and the energy of the reservoir is $E-\epsilon_i$. Now ...
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From emission probability to black body spectrum. (Hawking radiation as tunnelling.)

In the 1999 paper by Wilczek & Parikh (and in many subsequent papers) the "emission rate" (really the probability of tunneling) of a particle with energy $\omega$ from a static black hole (BH) is ...
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Would it be appropriate to say that in presence of Non-Conservative Forces, the Entropy of System Always Increases?

Would it be appropriate to say that in Presence of Non Conservative Forces the Entropy of a system would always increase? Can we relate Non-Conservative Forces and Entropy in some way? I intuitively ...
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2answers
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Why are the Lyapunov and Lindeberg Central Limit Theorem conditions often satisfied in the real world?

Some background for the question. I've been trying to understand why so many things have a Gaussian Distribution. There are a lot of questions about this on StackExchange but none of them were ...
0
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1answer
44 views

Why is potential energy zero in this calculation of partition function?

In section 7.2 of Rief's "Fundamentals of Statistical and Thermal physics". While calculating the partition function for ideal gas he writes: $$ \begin{array}{l} \displaystyle{Z' = \int{ \...
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1answer
311 views

Do there exist phases of matter where the order parameter space is non-orientable?

For example, are there order parameter space that is homeomorphic to a Klein bottle?
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2answers
58 views

Book(s) fills the gap from introductory thermo to nonequilibrium thermo/stat mech for self-taught student?

I know this is a big question. But as a graduate student, my research is somehow related to nonequilibrium thermodynamics/statistical mechanics. TBH, I hate how some research treat this subject like a ...
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2answers
76 views

What determines the timescale for fluctuations in the electromagnetic field from a light source?

Let's say you place an electric field meter some distance from a light bulb. As a function of time the output of the meter would be $\mathbf{E}(t)$. I would guess that the electric field will be some ...
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1answer
33 views

One dimensional paramagnet

I have a lattice model consisting of $N$ spins $s_{j}$ which can take the values $s_{j}=\pm1$. The spins are considered to be non interacting. The probability for a spin spin to be 1 is p and the ...
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1answer
47 views

How to find density of states in harmonic oscillator?

Density of state should be number of states per volume .Why weThe take derivative of "number of states " with respect to energy to get density of states ?
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1answer
38 views

Why does Critical Points have fluctuations on all scales (Infinite correlation length?

I have been studying statistical field theory for a while and I still haven't found a physical explanation for this question. Every answer seems to be kind of circular. Basically something like this: "...
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1answer
39 views

The average velocity of a particle

The Maxwell distribution of velocities is: $$p (v) = (\frac{m}{2\pi K_b T})^{\frac{3}{2}} e^{\frac{-mv^2}{2 k_b T}}$$ I want to understand how to obtain the average value of the velocity. The ...
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2answers
72 views

How to handle bra-ket in logarithm?

$\newcommand{\ket}[1]{\left| #1 \right>}$ $\newcommand{\bra}[1]{\left< #1 \right|}$ Say the following two equations: $$ S = - k_B \text{Tr} (\rho \ln \rho) $$ $$ \rho = \sum _\epsilon \ket{\...
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1answer
45 views

Computing the average energy and specific heat at constant volume

Consider an Einstein solid. Each oscillator has quantized energy $E = n\hbar\omega$, where $n \geq 0$ is an integer. How can I compute the average energy and the specific heat at constant volume of an ...
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1answer
44 views

Bose Einstein Condensation in Grand canonical ensemble

Why we develop formalism of Bose Einstein Condensation in framework of grand canonical ensemble ?