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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Marginal characteristic function from a multivariate charateristic function

[I am posting this question here and not in Math Stackexchange because I will be using conventions as they are usually used in statistical physics (especially the notation is more involved in pure ...
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What are many-body potentials? Why are they important?

I have been studying molecular dynamics and statistical mechanics, and I have been running into this term called "many-body" forces. I have been reading this paper: https://doi.org/10.1080/...
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Doubts in the proof of existence of empirical temperature as a result of zeroth law of thermodynamics

I am studying the mathematical proof of existence of empirical temperature online. I have studied the same in the post Proving the existence of temperature from zeroth law in the MIT OCW notes . ...
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Single-particle states in statistical distributions and the significance of Probability Distribution Function

This is a very basic doubt from Statistical Mechanics which I haven't been able to resolve yet. Suppose we take into account only single-particle states for a system, and we wish to look at the ...
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Are there sub-exponential local complex partition functions?

Consider an arbitrary local, translation-invariant, classical statistical lattice model such as the Ising or Potts model. The partition function $Z$ is a sum over products of local Boltzmann weights, ...
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Common subsystem-bath interaction operators?

Im new to the field of quantum open systems and I wanted to know what are the most common operators for describing the subsystem-bath interaction. To narrow down a bit my question, say we have as a ...
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Fraunhofer lines and kirchhoff's law of radiation

I am a high school student and I am very confused in a topic which is related to sun's spectrum, My teacher told me that the dark lines which we see in sun's spectrum are called fraunhofer lines and ...
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What happens to the Fermi level of a metal when a dc electric field is applied?

For metals, the Fermi level $E_F$, which is also equal to the chemical potential $\mu$ at absolute zero of temperature, is the highest occupied energy level at $T=0$. In terms of the electron number ...
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Is it possible to calculate how far vapor particles go?

Is it possible to calculate how far vapor particles go? I'm curious about vape pens and how far their smell goes. I thought if I think about it in terms of particles maybe I can come up with something....
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Is there any zero-order phase transition in nature?

Theoretically, a finite jump in the free energy phase diagrams can naturally be called a zeroth-order phase transition according to the Ehrenfest classification. We always hear about the first- and ...
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A Big Problem in Landau's Statistical Mechanics Book

This question regards: Landau; Course of Theoretical Physics, Volume 5, Statistical Physics, Part 1, Third edition, revised and enlarged. In Chapter IV, section 37 the objective is to derive the ...
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Average number of electrons per atom in a given state, given a thermal distribution

Im struggling with a question that I saw online. It is the following: An atom has 2 electron-states (which are not degenerate) with energies $ \varepsilon_{1},\varepsilon_{2} $. In the room ...
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How is entropy increase in an isolated thermodynamic system consistent with the unitary invariance of Von Neumann entropy?

I was reading this StackExchange answer, which I will briefly summarize here. There is a room that is at a fixed temperature and is isolated from the environment. There is a block of ice inside the ...
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Quantum relaxation to equilibrium?

Source and context: Im reading “The Theory of Quantum Open Systems” by Breuer and Petruccione. As an application of the just derived Lindblad equation for the dynamics of the reduced density matrix $\...
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Why only the ground state term in BE gases?

Usually most texts in statistical physics (e.g. Landau), and many papers I found on the subject, deal with Bose-Einstein statistics in the following manner: they use an approximation for which the ...
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I can't track too many particle ==> Statistical Mechanics OR Quantum uncertainity ==> Statistical Mechanics

I'm sort of confused with these two ideas. We start by saying that the trajectory of the particle can be described by giving the phase space coordinate $(q,p)$. So Given the initial condition, I can ...
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Are electrons in a p n junction delocalized across the junction?

When discussing p n junctions, it is usually said that electrons diffuse from the n side to the p side, hence creating a separation of charges. However, we know that electrons in a lattice are ...
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“Secular” approximation in the Lindblad equation for an open quantum system

Some context: I am deriving the Lindblad equation following “The Theory of Open Quantum Systems” by Breuer and Petruccione (somebody transcribed the section I am reading in this link). My question: I ...
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Problem regarding mean values of variations in statistical mechanics

Take a look at this previous question; in the answer by Julian Helfferich we arrive at the following formula for an infinitesimal change of entropy: $$dS = \frac{1}{T}\sum_i d(E_ip_i) - d(E_i)p_i$$ ...
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Average distance over all possible angles

I am working on an ideal chain problem in 3D. It is discretized, N links each with length l, and it is subject to a constant force in the +x direction. There are many solutions online, and I don't ...
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How to fully understand the Definition of Entropy?

In the context of Thermodynamics and Statistical Mechanics we encounter, basically, three different definition of entropy: First definition: Consider an isolated macroscopic system, it has a ...
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Chaikin and Lubensky exercise: upper critical dimension of anisotropic Lifshitz potential

Exercise from Chaikin and Lubensky: Chapter 5 Calculate the upper critical dimension $d_c$ for the following critical points: b) The $(d,m)$ Lifshitz point described by the Hamiltonian $$H = \int d^...
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General counting of microstates for a specific energy. Indistinguishable vs distinguishable

I have the following problem: I have two systems $A$ and $B$, each one with 5 distinguishable particles, $N_A=N_B=5$. The systems have infinite energy levels with energy $E_m = m \cdot \epsilon$ with ...
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Does appearence of $\hbar$ implies the role of quantum mechanics?

I'm currently confused with the uncertainty in statistical mechanics and in quantum mechanics. As I understand, If roll a die, then you can get an outcome from the sample space $$\mathcal{S}=\{1,2,3,4,...
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Density of states, Solid state physics

Im reading about Somerfeld's model, and under the assumptions of periodic boundary conditions for a particle in a box, the density of states in my book defined to be $$ g\left(\varepsilon\right)=\frac{...
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What is the definitive evidence we have that quantum theory's probabilistic nature is physical and not epistemic?

For example does superposition or wavefunction really occur in the physical quantum world or is it only a property of the quantum theory's framework and formalism to help us to make accurate ...
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How does the pressure change in the following scenario

Suppose there are n molecules of gas in a cylindrical container fitted with a piston. The force that was needed to keep the piston at a certain height depends upon the no. of molecules colliding with ...
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Extracting “useful” work from a free expansion?

Suppose we have an adiabatic and isothermal free expansion of an ideal gas at temperature $T_0$ so that the gas incurs an increase in entropy $$\Delta S_{sys}=Nk_B \ln\left(\frac{V_f}{V_i}\right)>0$...
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Helmholtz Free energy: Examples of constant temperature and volume processes?

The concept of Helmholtz free energy has been giving me a lot of issues and I think the main reason for this is that I do not really have many concrete examples of physical phenomena that I can think ...
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The definition of Spontaneous in thermodynamics?

The definition of spontaneous is often briefly glossed over in most of the thermodynamics texts that I own. Peter Atkins in Physical chemistry defines spontaneous as follows Some things happen ...
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Partition Function of a Hydrogen Atom

I'm trying to figure out how to write the partition function as a function of temperature for just a single hydrogen atom with a bound electron. I know the Energies will be $$E_n=\frac{-13.6eV}{n^2}$$ ...
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Memory effects in open quantum systems: Markovian approximation question

Setting: Quantum system $S$ composed of a subsystem of interest $A$ and a subsystem acting as the environment $B$ such that $S=A\cup B$. System $S$ is described by density matrix $\rho$ whereas ...
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When is molecular chaos dynamical chaos?

It is very common to have uncorrelated velocities in chaotic dynamical systems. Yet, we should be wary in equating the two quite different meanings of chaos. Instead of matching dynamical chaos to ...
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MOSFET current at very low temperature (4K)

I am trying to model the current at low T. To do so I need to compute the integral of fermi dirac statistics multiplied by the conductivity, with respect to energy: My problem is that I cannot ...
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Thermal velocity vs speed of sound

Is it possible that for a gas, speed of sound$>$thermal velocity? I cannot imagine it since, for the ideal gas limit, the interactions solely rely on the bouncing back of molecules
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More reversible heat required to do the same work as an irreversible process?

Suppose we have the reversible isothermal process from state a to b as shown below: The work done by the system is simply equal to the area under the curve. The fact that this process is isothermal ...
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2 dimensional Ising model: How do we visualize the Hamiltonian for interacting spins, but with no external magnetic field?

Suppose we don't have any external magnetic field, so that the Hamiltonian is given by $H=-J\sum_{i,j}s_is_j$. If we have an $n\times n$ 2D lattice of spins. Then does the $H$ correspond to one whole ...
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Plasma physics when plasma conditions don't apply

I've started to look into the field of plasma physics and I saw that not any gas of charges can be considered a plasma. There are 3 main conditions for something to be considered a plasma if I ...
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Does a broken symmetry lead to the second law of Thermodynamics? (Noether's Theorem)

So, Noether's Theorem states that for every conservation law, there is a corresponding symmetry. If that symmetry is broken, the conservation law no longer stands. Can this logic be applied to entropy?...
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Can the number of microscopic configurations of a gas be expressed in terms of its temperature?

I'm new to the field of thermodynamics, but have been thinking about it recently, and wondered whether this statement has any truth in it. I doubt it is true, however I need to see whereabouts the ...
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Why use partial pressure, not chemical potential, to describe vapor phenomena?

I have been looking up the basics on evaporation, boiling, humidity, and the dew point, in order to try to understand these everyday concepts with precise physics. I have noticed that "vapor ...
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Thermodynamics and the state postulate: should it be a Fourth Law?

The state postulate is as follows: The state of a simple compressible system is completely specified by two independent, intensive properties. My first question is whether there is any justification ...
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The Joule-Thomson Effect vs Adiabatic cooling

When an ideal gas in an adiabatic cylinder undergoes expansion, adiabatic cooling occurs and the temperature of the gas drops. This occurs because some of the gas's internal energy is used up as the ...
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Thermodynamic limit for an ideal gas

In S. Salinas, Introduction to Statistical Physics (Springer, 2001), the author states (p. 68):"...the thermodynamical limit is essential to allow the connection between the average values of ...
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How do we use ergodic theory in physics?

I am Mathematics student taking a graduate Ergodic Theory class. We are going over a lot of mathematical theory, but I would like to understand (at least at a superficial level) the connection with ...
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Estimating Debye and Fermi temperatures

Given the sketch of the behavior of the heat capacity against the temperature, How do I estimate the Debye and Fermi temperatures? Thank you in advance
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What is the velocity in the Langevin equation?

The Langevin equation is a stochastic differential equation for the velocity of one degree of freedom performing Brownian motion. It is supposed to describe the motion of a big particle at a much ...
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Derivation of the current density for the free electron gas

I have a few doubts about the study of the free electron gas model: We know that the current density expression may be calculated as follows J(r,t)=$\frac{\hbar}{2mi}(\psi^{*}\nabla\psi -\psi\nabla\...
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Equilibrium between two or more system

A system is said to be in a state of equilibrium if the thermodynamical variables are time-independent. Now, What it meant for two or more systems to be in thermodynamical equilibrium? In Kardar's ...
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On including conserved charges when maximizing entropy of a statistical system

One approach towards constructing the probability of a state in a system is by maximizing the constrained entropy $$ S[p(n);\alpha,\beta] = -\sum_np(n)\log(p(n))\;+\; \alpha\big(\sum_nE_np(n) -\langle ...

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