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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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Canonical and grand canonical formalism at zero temperature

How do we find expectation value of any operator at zero temperature in quantum statistical mechanics formalism? Expectation value of any operator $\hat{O}$ is given as $$\langle \hat{O}\rangle = Z^{-...
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63 views

How to interpret phase diagrams?

I find quite difficult to interpret phase diagrams in general, for example I see people discuss them along the following lines: Here we see the coexistence line between liquid-solid phases.. a ...
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19 views

BKT transition: nature of topological transition

BKT-transition is one of the most well-known topological transition in $O(2)$ model.But I misunderstand the physical interpratation of this transition. I started from the low-temperature expansion of ...
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43 views

Renormalization Group Decimation

I'm trying to understand how the renormalization group theory works. In the image shown on the bottom we coarse grain by decimation in the 2D Ising model. We split the lattice in two sets, $s^{\sim}$ ...
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2answers
34 views

In a NVE ensemble, can a particle access a state with a higher energy than the constraint?

I should start with saying that I know the answer is obviously no but I am trying to make sense of it mathematically. Let's say, for example, that each particle of a system with total energy $E$ in a ...
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1answer
52 views

Quantum statistical mechanics formalism

How do we solve a Hamiltonian written in second quantization by using quantum statistical formalism? For example, the following Hamiltonian $$H = \sum_{i=1}^L c_{i+1}^\dagger c_i + h.c.$$ I have ...
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1answer
30 views

Accurate derivation of electron-phonon scattering contribution to metal resistivity

My lecture derived the expression for this contribtuin using the collision integral approach but I missed lot of details. He considers the lowest order correction to distribution function $f=f_0+\...
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26 views

Hamiltonian is given of ising model in picture calculate the quantities [on hold]

How to solve this problem please help and help enter image description here
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1answer
40 views

Dealing with thermodynamic processes in Statistical Mechanics

I have recently started studying Statistical Mechanics, and through my study of Classical Statistical Mechanics, I have studied how do we write distribution functions for equilibrium systems which can ...
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51 views

Entropy and temperature of small systems

I've been struggling for a while now in understanding the concept of entropy as a function of the internal energy. Textbooks typically call $g(E)$ the number of microstates compatible with an energy $...
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20 views

The equality of fermi energy and chemical potential at high temperatures

Through the duration of my equilibrium statistical mechanics course, I came across a point in ideal Fermi systems where the Fermi energy was approximated as chemical potential when we derive the ...
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2answers
71 views

Why can many distinguishable particles be in the same quantum state?

In Boltzmann distribution, it says we have no restrictions on how many particles there are in a same quantum state. BUT the contradiction is: Distinguishable particles mean we think their wave ...
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1answer
108 views

Why is the $k$-space in multiples of $2\pi/L$?

So when you find the solution to the Schrödinger equation you get that the wave function can have $k=n\pi/L$, $n=1, 2,3 \dots $ The problem I have is that when calculating the density of states of a ...
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1answer
45 views

Why is the thermodynamic definition of temperature only valid for large systems?

I remember my thermal physics instructor telling me that the definition of temperature, $$\frac{1}{T} \equiv \left(\frac{\partial S}{\partial U}\right)_{N, V},$$ is only valid for (sufficiently) large ...
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27 views

Finite current in equilibrium in tilted Dirac Spectrum

I am calculting current for Dirac semimetal with energy dispersion $\varepsilon(s,p)= v_{T}^{x}p_{x}+sv_{F}|p|.$ Here $s=\pm$ refers to conduction and valence bands. The velocity for above spectrum ...
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1answer
50 views

Equation of state of Lennard-Jones spheres

One way of accounting for van der Waals interactions in fluids is to use the Lennard-Jones potential [*], which has a repulsive term that dominates at short distances which mimics the hard-core ...
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1answer
49 views

$N$ Independent Oscillators Entropy and Number of States

I have been working on building intuition and experience with solving problems related to the microcanonical ensemble, but I haven't been able to solve the following problem: If you have $N$ ...
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24 views

Microcanonical Ensemble Entropy of System with Degeneracy

So, I have been trying to find the entropy of the following system as a function of energy: the system consists of N atoms, N much greater than 1. Each atom has three energy levels: the ground state ...
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1answer
23 views

Number of particles needed when equilibrium energy is given [closed]

N particles obeying classical statistics are distributed among three states having $\epsilon_0=0$, $\epsilon_1=k_BT$ and $\epsilon_2=2k_BT$. If the total equilibrium energy of the system is $1000k_BT$,...
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19 views

Jeans equation for a fully isotropic velocity dispersion tensor

Given a known spherically symmetric gravitational potential, $\Phi(r)$, I need to calculate the value of the velocity dispersion tensor $\sigma_{rr}$. To calculate $\sigma_{rr}$ I am using Jean's ...
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1answer
42 views

Why do purely harmonic interatomic interactions result in infinite thermal conductivity?

When interatomic interactions are purely harmonic, normal modes cannot interact, and therefore no phonon scattering occurs, thus resulting in infinite thermal conductivity. But why is anharmonicity ...
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26 views

How does quantum statistical mechanics explain thermal conductivity?

I am unsure how statistical mechanics can be used to explain the dynamics of so called non-mechanical properties, e.g. Temperature and Entropy (as opposed to say pressure or spatial diffusion). In ...
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0answers
21 views

Why are magnon/phonon/photon velocities linear response quantities?

On page 2 of this paper, Onsager reciprocal relation for linear response is introduced as $$K_{AB}(q,\omega,B)=\epsilon_A\epsilon_B K_{BA}(-q,\omega,-B)$$ where $\epsilon_A,\epsilon_B=\pm1$ specifies ...
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1answer
79 views

Why does quantum mechanics become unnecessary at sufficiently high temperatures?

In my statistical mechanics intro class, we are taught that at sufficiently high temperatures, the quantum treatment of things becomes unnecessary. Why is this? Can this be shown using certain ...
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26 views

The importance of phase when defining fermions and bosons

In my lecture on indistinguishable particles, my lecturer is trying to illustrate to me the notion of particles being indistinguishable when considering that when we swap two particles in a box the ...
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2answers
26 views

Physical implications of no. Of microstate

I was studying the Statistical mechanics and what I have understood is that if there is a large number of particles in a system , and if we want to study the system then we have to calculate the ...
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4answers
3k views

What force causes entropy to increase?

What force causes entropy to increase? I realize that the second law of thermodynamics requires the entropy of a system to increase over time. For example, gas stored in a canister, if opened ...
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1answer
56 views

Integrability of a non-integrable quantum spin model at critical point

Is it right, that non-integrable quantum spin models in one dimension become integrable at their critical points? Or do they stay nonintegrable at the critical point also? Are there any examples known?...
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3answers
71 views

Deriving the canonical ensemble from the microcanonical ensemble: Why expand the logarithm of the probability and not some other function

Several other posts on the derivation of the canonical ensemble give an explanation in terms of considering a subsystem of a larger microcanonical ensemble see How is the distribution probability in ...
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27 views

Bose Einstein Condensation

We all know that the Bose Einstein distribution formula is $n=g_i/(e^{α+βi}-1).$ Where alpha is the derivative of constant number of particles that is supposed to be zero. But while studding Bose ...
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0answers
31 views

Why is finite temperature many-body perturbation theory computed in the grand canonical ensemble?

Why does virtually every textbook and paper treat many-body perturbation theory at finite temperature in the grand canonical ensemble? Is it not possible to formulate a canonical theory where all ...
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0answers
42 views

Fluctuation-Dissipation Theorem in the Keldysh Formalism

In Kamenev's book Field Theory of Non-Equilibrium Systems (he also has lecture notes online here, which contains the relevant statement on pg. 17), he states that the following equation $$G^K(\epsilon)...
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2answers
73 views

Why should the equal a priori states be eigenstates of the Hamiltonian?

When I have was taught quantum statistical mechanics, the course derived the canonical ensemble by assuming that eigenstates of the total system Hamiltonian have equal a priori probabilities, and then ...
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1answer
53 views

Whats the function of density of states equation?

There was a question in Statistical Mechanics 3rd ed by RK Pathria and PD Beale, section 3.8, asking to show that the harmonic oscillator obeys the equipartition theorem. It was well proven. But at ...
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1answer
25 views

Nose-Hoover thermostat: real-time average problem?

Many articles as well as the following presentation MD Ensembles and Thermostats, page 23, claim that the principle of the nose-hoover thermostat is "removing the real-time average"-problem. I can ...
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26 views

Expand Helmholtz energy to fourth order in m (magnetisation)

The energy of a system of spins is given by: $$H=\frac{1}2\sum_{i,j=1}(J\sigma_i\sigma_j-h\sum\sigma_i)$$ I found that we can rewrite in terms of magnetisation, $m=\sum\sigma_i/N$: $$E=-N(\frac{Jm^2}...
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1answer
44 views

Deriving $\Omega$ for distinguishable particles

For atoms that are fixed in a crystal with the following assumptions: There are quantised energy levels with energy $E_i$ for each atom. Each state has a distinct energy $E_i$. Such that.. $$N = \...
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1answer
32 views

Structure factor in a homogenous system

I want to calculate the structure factor for a homogenous system. The system that I am dealing with is the results of a Vicsek type model simulation. The structure factor is defined as : $$S(q) = \...
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0answers
32 views

Canonical ensemble: average value

I am having trouble with the following physics question. I have a quantum system of N distinguishable particles, whose energy is given by: $\varepsilon_{n}=n\Delta\qquad n=0,...,n_{0}$ I don't know ...
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20 views

Critical parameter for 1D quantum system corresponding to $T_c$ of 2D Classical model

Utilizing the fact that there is a correspondence between a $d$ dimensional quantum system and a $d+1$ dimensional classical system (c.f. Trotter Decomposition), my question regards what the critical ...
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2answers
69 views

Where does the energy for osmosis come from? [duplicate]

A thought experiment: A U-shaped tube with semi-permeable membrane at the base. The tube is completely thermally isolated from its surroundings. The liquid (solvent) is at some temperature $T$. ...
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1answer
41 views

In the Rayleigh–Jeans Law, why polarization is two?

When we count mode of wave in cavity radiator, why the "2" is multiplied by polarization? Polarization has two base, but the number of polarization state maybe infinity and continues. Linear ...
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1answer
25 views

Change of entropy of mixing in terms mass ratio

I am stuck on this problem. Suppose we are mixing substances $a$ and $b$, we have $$\Delta S_{mixing} = -Rn_a \ln (\frac{n_a}{n_a + n_b})-Rn_b \ln (\frac{n_b}{n_a + n_b})$$ and we are told to ...
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1answer
33 views

Intrinsic probability in correct Boltzmann counting

Why $\lambda_i$ can be set to unity? What it means? That picture comes from Huang's book.
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3answers
99 views

How can temperature be defined as it is if entropy isn't a function of energy?

I've been taught that for large systems, the temperature of a system is defined as $$\frac{1}{T} \equiv \left(\frac{\partial S}{\partial U}\right)_{N, V}.$$ I have a problem with this definition, ...
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1answer
48 views

Classical Molecular Dynamics Simulations: Who cares about partition functions?

In most introductory textbooks on molecular dynamics simulations the authors usually tell us how the connection between micro and macro is established using the partition function. Knowing the ...
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0answers
42 views

Why do the $C_v$ of gapless systems have a power law behaviour?

The functional dependence of the heat capacity $C_v$ of systems with gapless excitations (e.g., lattice with acoustic phonons, Heisenberg ferromagnet with spin waves etc) is like a power law $$C_v\sim ...
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3answers
53 views

Reversible reaction for Entropy Change of Surroundings

When learning about Entropy in my introductory lecture, I learnt that in basic terms, entropy can be spoken about as $$dS = \frac{dq_{rev}}{T}$$ and the lecturer mentioned that as Entropy $S$ and $T$...
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Textbooks about spin-glasses for beginners

I am a Ph.D. student in Physics attending my second year. I would like to ask you whether you know any good textbook about spin-glasses (and physics of complex systems in general) for beginners. ...
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3answers
2k views

Ambiguity in the definition of entropy

The entropy $S$ of a system is defined as $$S = k\ln \Omega.$$ What precisely is $\Omega$? It refers to "the number of microstates" of the system, but is this the number of all accessible microstates ...