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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What does integrating the probability density function over all phase space gives us?

For a system of N-3D particles, we have 6N D.O.F and therefore a 6N dimensional phase space. I know that one point in phase space represents a possible state of the system. I also understand that a ...
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Pair correlation function for non-interacting spinless bosons

While trying to solve a second quantization exercise regarding a bosonic gas, I've been having trouble trying to understand the $(1-\delta_{\text{pq}})$ term in the decomposition of the following ...
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In which cases you use free energy to calculate pressure?

From thermodynamic identity you get: $$\left(\frac{\partial U}{\partial V}\right)_{S,N} = -P$$ But with Helmholtz free energy $F = U-TS$, we can also get pressure from this equation: $$\left(\frac{\...
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Approximation for average thermal photon number

I am currently reading the book Introductory Quantum Optics by Gerry and Knight and I am having trouble understanding an approximation they make. In the chapter on thermal fields they derive the ...
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Apparent paradox in statistical mechanics

I can't understand why the likelihood of a particle to be in state $\epsilon_i$ in a canonical ensemble, does not depend upon the number of particles in that state. The probability that the a single ...
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What is the relation between the partition function from Stat. Mech. And the Path Integral? [duplicate]

Beside the fact that they look identical when you take imaginary time in the path integral formulation. I understand we doing statistics and we are just integrating over all states with a relative ...
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41 views

Maxwell-Boltzmann distribution as the most probable

In Statistical Physics: An Introduction, by Daijiro Yoshioka, there's a neat derivation of the Maxwell-Boltzmann distribution as the most probable distribution of an Ideal Gas. He uses the method of ...
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Question regarding expectation value of energy and gibbs factor

In lecture we introduced the average energy for a single particle as $$E = \langle \varepsilon \rangle = \frac{\int d \varepsilon g(\varepsilon) e^{- \beta \varepsilon} \varepsilon}{Z_1}$$ where $Z_1 =...
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What would be the proper distribution to model the number of particles in a state in canonical ensemble

Suppose my system has $N$ particles, and I want to find a distribution for $n_i$, the number of particles in the $\epsilon_i$ energy state. What I do know is the boltzmann probability, which tells me ...
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51 views

Does the second law of thermodynamics hold under non-equilibrium conditions? [duplicate]

The question is just that: whether the second law of thermodynamics is always valid under non-equilinrium conditions? The origin of life involves several chemical reactions that are thermodynamically ...
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Confusion regarding Gibbs' paradox

I am trying to understand the following commentary I found in Wikipedia about this paradox: Now a door in the container wall is opened to allow the gas particles to mix between the containers. No ...
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Is Gibbs/Boltzmann probability the 'true' probability of a particle being in a particular state in the canonical ensemble

Based on the classical interpretation of probability, the probability for a single particle to be in the $i$th energy state, in an $N$ particle system, should be given by the number of particles in ...
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Latent Heat in the Ising Model

I am confused about the (non)-existence of latent heat in the first order phase transition of the Ising model. Most textbooks talk about latent heat as a signature sign of a first order transition, ...
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Entropy in a reversible process

According to 2nd law of thermodynamics entropy never decreases, it's either zero or bigger. The problem with the definition is that it doesn't specify WHICH entropy never decreases? Of the system that ...
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Can the lifetime of an electron in the conduction band be calculated?

We know that in a semiconductor in every instant of time some electrons get excited from valence band to conduction band and some electrons are deexcited from conduction band to valence band. In this ...
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Continuous phase transition with no finite critical exponent

I am working with a model in which the energy density as a function of chemical potential $\mu$ and density $n$ is given by $$E = (e^{-1/n}-\mu)n$$ in appropriate units. This model has a phase ...
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Expectation value of energy in a canonical ensemble

Suppose we have a canonical ensemble system of $N$ particles, and $k$ single-particle-energy levels labeled by $\epsilon_i$. The energy of the different microstates of the entire system is given by $\...
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Is the total energy of a canonical ensemble system of $N$ particles, with single-particle energy levels given by $\epsilon_i$ fixed?

Is the total energy of a canonical ensemble system of $N$ particles, with single-particle energy levels given by $\epsilon_i$ fixed ? We know the total energy of the system is given by : $$E=\sum_{i} ...
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Interpretation of probability in Statistical Mechanics

In statistical mechanics, in particular the canonical ensemble, the probability of the system to have a particular state is given by : $$P_i=\frac{e^{-\beta E}}{Z}$$ Here $Z$ is the partition function ...
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Exact heat capacity of the 2-dimensional Ising model

The following is a section from the book Newman, M., and G. Barkema. "Monte carlo methods in statistical physics" New York, USA (1999). and then: From those two quotes, it seems that there ...
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Confusion regarding the use of partition function

Suppose we have a system filled with $N$ particles. There are $k$ energy levels in this system, labeled by $\epsilon_i$, each with a degeneracy of $g_i$. Let us imagine $n_j$ particles out of these $N$...
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Confusion about using single particle or $N$ particle partition function in Boltzmann probability in canonical ensemble

Suppose we have a canonical ensemble, where $N$ particles have been divided among $\epsilon_i$ energy levels, each with degeneracy $g_i$. The partition function for a single particle is given by : $$...
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Rewriting of a occupation based Hamiltonian to an spin based Ising Hamiltonian

I run in to the following problem of rewriting a hamiltonian derived in an earlier question to an Ising hamiltonian. (b) Identify your result in (a) with the Hamiltonian and the partition function of ...
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Physics book recommendation for a mathematician studying statistical mechanics [duplicate]

I just started my PhD in statistics and I am very likely to defend my thesis on statistical mechanics. Having said that, as I am a mathematician by education, I do not have a strong background on ...
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What is the relation between joint measurability and common refinement (pure state decomposition) of density operators?

Here page 13, the author states "...just as two quantum observables are often not jointly measurable, two decompositions of mixed states often have no common refinement (Actually, in the ...
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Langevin Paramagnetism for dipoles rotating in 2D

The energy for dipoles in a magnetic field can be described by $$H = - \mathbf{m} \cdot \mathbf{B}.$$ What I did for an exercise was integrate the partition function for the case where I allow ...
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Microstates/ Eigenstates of Hamiltonian and multiplicity

I have 2 specific question regarding micro-states, entropy and the relation with the hamiltonian. If we observe a closed system, then the microstates of it are eigenstates of the hamiltonian. Is ...
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Form of second-order expansion terms in classical linear response theory

I am studying the linear response theory from this [1] reference which basically tries to calculate how a system in equilibrium responds when a force is applied. I understand for the first-order we ...
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Interpretation of entropy flow

Recently I have dealt with the following schematic equation (I avoid technical details in order to concentrate on conceptual things): $$\frac{dw}{dt} = \frac{\partial S}{\partial w}, \tag{*}$$ where $...
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What is (local) pressure within a gas on the microscopic level?

Fluid dynamics view Fluid dynamics describes liquids and gases in terms of local pressure (among other variables), which varies from a point to point. While mathematically the concept is well-defined (...
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Kinetic energy of an electron inside the sun [closed]

I have to tell if an average proton or electron at 15 million degrees kelvin be treated by relativistic mechanics. The criterion is that the kinetic energy differ from the classical kinetic energy by ...
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Is it meaningful to compare radial distribution functions between different species?

The radial distribution function corresponding to $g_{OO}(r)$ in water is moved in a little more than the $g(r)$ for argon, and the peaks corresponding to the second, and beyond, coordination spheres ...
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How is linear momentum conserved in a cooling gas?

Consider a closed cylindrical tank with no external forces acting on it which is filled with gas which is cooling due to radiative heat loss to the surroundings. Consider a smaller tank at a higher ...
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Standard Relation in terms of partition function and beta

$$(\Delta E)^2 = \frac{\partial^2 \ln{Z}}{\partial \beta^2} \tag{2.29}$$ Shortly, how can I obtain this relation? I found this relation from Franz Mandl Statistical Physics The following are related ...
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Averaging SYK models and the disappearance of the density matrix

In A strongly correlated metal built from Sachdev-Ye-Kitaev models by Song et al. they wish to calculate the generating function for a system with quenched disorder. In the Keldysh formalism, this ...
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Reversibility of the arrow of time

I often read in physics vulgarisation books about how paradoxal it is that the time seems to go only one direction, as entropy grows with time, and nobody has ever seen a broken cup repair itself and ...
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Saturated vapor and liquid densities (VLC curve) from Helmholtz free energy equation of state

I have been reading about the statistical associating fluid theory (SAFT) which computes the Helmholtz free energy ($A$) from the SAFT EoS (a hell lot of equations) for a molecule of interest, and is ...
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Connection between the imaginary part of retarded correlation function and derivative of Fermi-Dirac distribution function

A two-particle retarded correlation function is (its derivation is not related to my question here) $$ C^R(\omega) = \sum_{kq}\bigg(f(\epsilon_k )-f(\epsilon_{k+q} )\bigg)\frac{1}{\omega+\epsilon_k-\...
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Quantum to classical mapping

I'm having troubles understanding precisely how the mapping from a quantum system to a classical one works. Let's say that I have a quantum system in $d$ dimensions with Hamiltonian $H$ at temperature ...
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43 views

Free energy landscape and probability distributions

I couldn't understand the concept of free energy landscape. Usually free energy is defined in the following way. $$F=-k_{B}T\ln Z=-k_{B}T\ln\sum e^{-\beta \epsilon}$$ where $\epsilon=\sum\frac{p_{i}^{...
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Derivation of Conditional Shannon Entropy [closed]

In the following question, I have found a few derivations for the conditional entropy, but the terms are in a different order. Show that Equation (5.11) comes from evaluating $$H(X|Y)=-\sum_{j=1}^N\...
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Entropy of $n$ particles in a $N \times N$ grid with neighbouring forbidden sites

Imagine an $N \times N$-grid on which $n$ particles should be distributed. In a non-interacting case (while each grid point can accommodate max 1 particle), the entropy is given by: $$S = k_{\mathrm B}...
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Partition function of 3D quantum harmonic oscillator

The following discussions are for isotropic quantum harmonic oscillators which have the energy eigenvalues as follows: $$E=\left(\sum_{i}^{N}n_i+\frac{N}{2}\right)\hbar \omega$$ where $N= $ number of ...
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Why at high energies, Fermi-Dirac and Bose-Einstein distribution behaves as Maxwell-Boltzman distribution? What is the physical explanantion?

I was searching for the reason that why at higher energies, the FD and BE distributions behave as MB distribution. In Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles by Eisberg and ...
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Why can we assign both energy and particle number to each state in grand canonical emsemble?

The system in grand canonical emsemble together with the surrounding reservoir is isolated, thus have conserved particle number $N$. However, the system itself only has fixed average $\langle N\rangle$...
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What is a correlation length? [duplicate]

What is a correlation length? I encountered this term in my space physics lecture, in the context of the "correlation length of the magnetic field magnitude," but I am not sure what does it ...
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Entropy-per-site model leading to "shapes" of generating function roots

This question is cross-posted from MathOverflow. Many statistical mechanics systems have well-defined entropy-per-site function $f(x)$ with respect to some control parameter $x$. This question is ...
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149 views

Legendre Transformation of Landau Free Energy

I am trying to get an intuition for the Legendre Transformation of a generic Landau free energy, e.g. for the Ising model with magnetization $m$ given by $$F(m) = \frac{a}{2} m^2 + \frac{b}{4} m^4 + \...
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Why Can Electrons be Modelled as Classical Spins?

Although electrons are spin $1/2$ particles described by the Pauli matrices, the Ising model treats electrons as classical spins. As a result, the Ising model does not describe anything physical, but ...
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Absence of Symmetry Breaking in 1D Ising Model--Continuum Version

I have seen arguments for why there is no symmetry breaking in the 1D Ising model--for example, using the transfer matrix method to explicitly solve the model, and another of energy-entropy arguments ...

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