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Questions tagged [boltzmann-equation]

DO NOT USE THIS TAG for Boltzmann's constant, Maxwell-Boltzmann distribution, Stefan-Boltzmann law & Boltzmann brains!

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How does the extra term and sign change come from in Harold Grad’s derivation of Boltzmann’s equation? [closed]

Harold Grad’s method of deriving the Boltzmann equation starts by integrating the Liouville equation: $$\int_{D_1}\left[{\frac{\partial F_N}{\partial t} +\sum_{i=1}^{N} {\vec{v}_i}.\frac{\partial F_N}{...
Jyotishraj Thoudam's user avatar
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Receipt for writing down the Boltzmann-equation for a given interaction

Suppose one has a gas consisting of two particles, which are known to primarily interact with each other through a specific interaction. To describe the behavior of the particle densities, I would ...
Luca D's user avatar
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3 answers
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Boltzmann distributions on atomic orbitals: infinite degeneracy?

The (unnormalized) Boltzmann probability distribution of states as a function of energy and temperature is given by $$P(\epsilon_i) \propto g_i\exp\left(\frac{-\epsilon_i}{k_BT}\right)$$ with $P(\...
ChangedMyName's user avatar
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Assumption of molecular chaos (confusion)

The molecular chaos means that the velocities of the two particles are uncorrelated before the collision. However, this is no longer the case after the collision, which is why the whole problem with ...
Raul E.'s user avatar
2 votes
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Is it possible to say that entropy would affect the charge transportation?

I noticed there are a few papers from Dr. Karuppuchamy Navamani, for example: Generalization on Entropy-Ruled Charge and Energy Transport for Organic Solids and Biomolecular Aggregates Theoretical ...
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1 answer
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Distribution of states of H$_2$ gas for a given temperature

How can I calculate the population distribution of vibrational and rotational states of H$_2$ gas, for any given temperature? I think the vibrational states are more important, since the rotational ...
Random_Astro_Student's user avatar
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Conservation Equations Used in Boltzmann Transport Equation: Kardar

I am reading through the derivation of the Boltzmann transport equation in the text, Statistical Mechanics of Particles by Kardar. I am unable to make sense of the conservation equations described. ...
fantasieImpropt2's user avatar
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1 answer
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Expressions for Entropy in the Canonical Ensemble

In the microcanonical ensemble, we have the standard Boltzmann expression for entropy: \begin{equation}\label{1} S = k_B\ln \Omega \end{equation} where $\Omega$ is the number of elements of the ...
Johnny Smith's user avatar
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How is Weiss mean field approximation actually maximising the partition function of Boltzmann's distribution?

Considering other mean field approximation (e.g. Max entropy approach or $<S_i> = m_i +\delta S_i$ , $\delta S_i \simeq0$), a common approach that I've seen is that of maximising the partition ...
zioperw's user avatar
2 votes
0 answers
44 views

Definition of temperature in Boltzmann transport theory

In kinetic theory, the local temperature in a fluid is defined in terms of the average thermal energy of the particles, as \begin{equation} \langle E_{th}\rangle= \int {d^3v}\frac{1}{2} m|\mathbf v-\...
Ramal Afrose's user avatar
5 votes
1 answer
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Acoustic finite-size effects of simulated fluids under periodic boundary conditions

Consider a fluid simulated in a finite box of specific size. An impulse to the fluid element at the center in a given direction is physically expected to propagate at the speed of sound and attenuate ...
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"Boltzmann" equation for radiation (Reheating period)

These equations are given in many papers as the "Boltzmann equations" (without derivation) governing the reheating period, where $\rho_\phi$ is the energy density of the decaying inflaton ...
Rajat Mondal's user avatar
1 vote
2 answers
143 views

Question about the Boltzmann Distribution for an ideal gas

The general statement of the Boltzmann distribution law is that for a system of $N$ particles, each having access to energy states $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_k$, the ratio the ...
Neel's user avatar
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Is there a way to express the collisionless boltzmann equation in terms of positions, velocities, times, without the distribution function?

Suppose I have data that represents a field of positions and velocities. If the distribution function (DF) for the data is $f(x,v,t)$, I know that the DF must obey $$\frac{\partial f}{\partial t} + \...
James Thiamin's user avatar
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Difference between thermodynamical, statistical, and dynamical equilibrium

I have two related questions concerning the difference between thermodynamic, statistical mechanical, and dynamical equilibrium. In particular, I am thinking about the statistical physics of galaxies, ...
FriedBarking's user avatar
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1 answer
115 views

A question on relaxation time approximation

I was learned that the form of collision term in relaxation time approximation is set to be: $$\left (\frac{\partial f}{\partial t}\right)_c=-\frac{f-f^0}{\tau}$$ in with $f^0$ is local equilibrium ...
Haiqin Tang's user avatar
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How to find the time derivative of temparature by solving the relativistic Boltzmann equation?

I am trying to find the temporal derivatives of the temperature by solving the relativistic Boltzmann equation. However, it didn't turn out as simple as I thought. The relativistic Boltzmann equation ...
cosmiccosmo's user avatar
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Verification of the second law of thermodynamics for liquids?

I have a pure math background and I am currently self-learning physics. To mathematically justify and understand the Second Law of Thermodynamics, mathematicians and physicists have studied the motion ...
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2 answers
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Why two systems with similar space of possible microstates have different entropy?

Boltzmann's entropy formula: $S=k_{\mathrm {B} }\ln \Omega$ where $\Omega$ is the number of real microstates corresponding to the gas's macrostate. Let's assume that we are talking about an ideal gas ...
Kaveh's user avatar
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Is nonlinearity a denser encoding of information?

At the microscopic level, an $n$-particle system in 3D can be described by the Liouville equation, which governs the evolution of the distribution function in a $6n$-dimensional phase space. Going ...
confusion's user avatar
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Boltzmann equation for Lennard-Jones particles?

In the Vlasov variant of Boltzmann equation, the Coulomb pair forces are included via the force term that appears as the prefactor of the derivative of the density with respect to velocity. In that ...
YoussefMabrouk's user avatar
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From Newtonian mechanics to Boltzmann (or statistical) mechanics

Classical mechanical systems observable on a dynamical scale are subject to Newton's laws. In this case, knowledge of the Hamiltonian allows us to minimize energy taking into account inertia. This ...
YoussefMabrouk's user avatar
1 vote
1 answer
48 views

Value of Inverse of Thermal Voltage

I am trying to solve the Poisson-Boltzmann Equations and have come about the constant $$ \beta = \frac{e}{K_BT} $$ which Wikipedia denotes as the inverse of Thermal voltage. Now this constant is ...
Abhinav Jha's user avatar
3 votes
1 answer
330 views

Issue deriving Fokker-Planck equation starting from Boltzmann's equation

I was trying to derive the Fokker-Planck equation starting from the Boltzmann's equation and I run into some issue while trying to do so. Starting from Boltzmann and using the notation $f \equiv f(x, ...
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3 votes
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How to find plasmon from Landau-Silin equation?

In David Pine's Theory Of Quantum Liquids: Normal Fermi Liquids, it's said that we can find charged Fermi liquid has plasmon modes easily from Eq. (3.40), replicated as follows: $$ (\boldsymbol{q} \...
jywu's user avatar
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Why are two assumptions for the approximation from the BE, FD to MB distribution and $T$ invariance in the Boltzmann equation reasonable?

As the title says, I can not understand whether the assumptions are reasonable. If an interaction $1+2\leftrightarrow 3+4$ is taken into account, the number variance in time is proportional to $$|\...
Jae Hoon Jeong's user avatar
12 votes
2 answers
824 views

Relaxation of the Boltzmann transport equation

My professor in kinetic gas theory said that when considering the Boltzmann Transport Equation (BCE) $$ \partial_tf + \frac{\vec{p}}{m}\cdot\nabla_{\vec{q}}f + \vec{F}\cdot\nabla_{\vec{p}}f = (\...
Tomas Noguera's user avatar
1 vote
0 answers
99 views

What should be the definition of a comoving frame in phase space?

In short, I think there are two types of comoving frame when talking about distribution function, since it is defined in phase space. Which one should be the real one? I suppose this question is ...
sherz's user avatar
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4 votes
1 answer
137 views

Hydrodynamic equation to Boltzmann's equation

How to get the four-velocity of a fluid in terms of its microscopic distribution function $f(x^{i},\vec{p})$? For the sake of initial simplicity, the fluid can be thought of to be single component. ...
SCh's user avatar
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How do you use Boltzmanns distribution law?

Considering a particle in an isothermal atmosphere: $ f \left( h \right) \delta \left( h \right) = A e^{\frac{-m g h}{k T}} \delta \left( h \right)$ where $A$ is the normalisation constant of the pdf. ...
thicccjk's user avatar
1 vote
1 answer
83 views

Boltzmann Equation Maximum Entropy Production Principle

I am currently reading this paper on the maximum entropy production principle (don't confuse it with the maximum entropy principle). Equation 2.1 (ommiting the term with the external force) is the ...
eeqesri's user avatar
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1 vote
1 answer
94 views

Maxwell Boltzmann distribution of speed [closed]

I am currently reading 'Concepts in Thermal Physics' and in its chapter 5 it is written that fraction of molecules traveling with speed between $v$ and $v+\mathrm{d}v$ corresponds to a spherical shell ...
Mr. Wayne's user avatar
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2 answers
155 views

Boltzmann distribution and probability of finding the system with specific energy

For sake of simplicity assume classical discrete systems. If we have a system ($\text{S}$) coupled to a reservoir ($\text{R}$), then a microstate of the combined (isolated with fixed energy $E$) ...
Antonios Sarikas's user avatar
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1 answer
221 views

Introductory text to the Boltzmann equation?

I'm searching for a good introductory text to the Boltzmann equation and how it gets applied in the relativistic case as well?
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0 answers
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Equilibrium carrier distribution appropriate to local temperature

In Chapter 13 of Ashcroft and Mermin there is a general discussion about the nonequilibrium distribution function under the relaxation time/semiclassical transport assumptions. One of the key axioms ...
EE18's user avatar
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1 vote
1 answer
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Clarify derivation: Why is the last term of (5.37) in Modern Cosmology by Dodelson and Schmidt zero?

I'm going through Modern Cosmology by Dodelson and Schmidt 2nd edition, and I'm stuck at (5.37) where they present the Boltzmann equation for dark matter and take the integral of all the terms over ...
Maximal Ideal's user avatar
2 votes
1 answer
117 views

Boltzmann Transport Equation existence and smoothness - Is it proved?

Currently, Navier-Stokes Equation, its solution's existence and smoothness is not well established, making the problem as one of famous Millennium Prize Problems. On the other hand, I noticed that ...
K.R.Park's user avatar
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2 votes
1 answer
30 views

How does one choose the moments used in MRT relaxation within the Lattice Boltzmann method?

I am trying to understand MRT relaxation in the lattice-Boltzmann method. The central equation in the lattice-Boltzmann method to simulate fluid flows is: $$f_i(x + c_i\Delta t, t + \Delta t) = f_i(x,...
tgv's user avatar
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1 answer
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Elastic Collision Point Masses / Hard Spheres:: Proof that Magnitude of Relative Velocity is Unchanged

Statement of the Problem On our way to the Boltzmann Collision integral, we consider the perfectly elastic collision of two point-masses with identical mass. The velocities prior collision are denoted ...
Dan Doe's user avatar
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0 answers
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Electron transport in fluids VS solids ("structure factor" VS "phonons")

Assume to have a neutral plasma consisting of some ions $X^+$ in a neutralizing gas of electrons $e^-$. Imagine that the $X^+$ can be in a gaseous (G), liquid (L) or solid (S) phase. We want to ...
Quillo's user avatar
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3 votes
0 answers
91 views

Does the Boltzmann equation reduce to Navier-Stokes equation? [duplicate]

Since the derivation of the Boltzmann Equation uses the molecular chaos assumption, it seems to me that it should not be valid for dense systems such as fluids. Now, according to Chapman-Enskog theory,...
Roshan Tom's user avatar
1 vote
0 answers
53 views

Modelling electrical conductivity in low-dimensional nanostructures

I know that the Boltzmann transport equation can be solved under the Relaxation Time Approximation (RTA) to obtain the electrical properties of materials. However, the parameter that I am interested ...
PBH's user avatar
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1 answer
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Transition from local equilibrium to global equilibrium in Boltzmann's equation

My question is about the evolution of a system from local equilibrium to global equilibrium. The system is described by a Boltzmann transport equation $$ \dfrac{\partial f}{\partial t}+\mathbf{v}\cdot\...
Jose Menendez's user avatar
1 vote
1 answer
579 views

What Assumptions Govern the Applicability of the Boltzmann Distribution?

In the book "Concepts in Thermal Physics" the Boltzmann distribution is derived with the following assumptions: There are two systems, one enormous heat reservoir and one comparatively ...
Connor's user avatar
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3 votes
0 answers
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How to reduce an integral in phase space to an one-dimensional form?

I've been trying for a very long time to show that the following integral: $$I_D=2{\displaystyle \int} \, {\displaystyle \prod_{i=1}^3} d \Pi_i \, (2\pi )^4\delta^4(p_H-p_L-p_R) |{\cal M}({e_L}^c e_R ...
RicardoP's user avatar
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-2 votes
1 answer
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Resampling step of population annealing

I am trying to have a better understanding of the population annealing algorithm. In the section POPULATION ANNEALING, the steps of the algorithms are described. In Step 2 (Resampling (Split/Remove) ...
Omar Shehab's user avatar
1 vote
1 answer
214 views

Boltzmann equation for photons in cosmology

I am trying to understand the derivation of the Boltzmann equation for photons given in Modern Cosmology (2nd Edition) by Scott Dodelson and Fabian Schmidt. In Eq. (5.4) they give the zeroth-order ...
Virgo's user avatar
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7 votes
3 answers
1k views

What is the difference between Liouville's theorem and the Boltzmann transport equation?

From what I understand, Liouville's theorem is about the probability density $\rho$ of an ensemble existing in a differential volume in phase space $d\mathbf{r}d\mathbf{p}$. So the statement for ...
megamence's user avatar
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0 votes
3 answers
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What is the actual meaning of velocity space is isotropic or density of velocity points is same everywhere?

I am studying Maxwell Boltzmann distribution law, In the derivation I came across a statement and I am not sure if my interpretation is right or wrong. Statement is "since velocity space has been ...
Nikhil Pathak's user avatar
3 votes
0 answers
56 views

Understanding $D \partial^2 P(x,v,t)/\partial x^2$ as a type of collision term

Using conservation of particles in a control volume in phase space (in one dimension with no sources of particles or external forces), one can derive the formal transport equation $$ \partial_t P(x,v,...
kevinkayaks's user avatar