text stringlengths 81 47k | source stringlengths 59 147 |
|---|---|
Question: <p>I was browsing my friends old notes and I came across the following problem that I am not sure if it's correct.</p>
<blockquote>
<p>Q. Prove that the molar entropy of <strong>CO</strong> of $0K$ would be $R \ln 2$.</p>
</blockquote>
<p>Here, it is considered that the thermodynamic probability of $N$ mo... | https://physics.stackexchange.com/questions/74972/what-molecule-would-have-molar-entropy-r-ln-2-at-0k |
Question: <p>First of all, I would like to apologize in advance if I make stupid mistakes. I am a mathematician and I am trying to apply the Boltzmann distribution to places where I am not sure if it is applicable (albeit I have no choice). </p>
<p>The situation is: I have a system which consists in a discrete line of... | https://physics.stackexchange.com/questions/78524/boltzmann-distribution-with-interaction-between-particles |
Question: <p>I am having problems in understanding the logic of this distribution:</p>
<p>$P(\Psi_{j})=\displaystyle\frac{e^{-E_{j}/kT}}{\displaystyle\sum_{j'}e^{-E_{j'}/kT}}$</p>
<p>The book I am studying use the case of a sample in contact with a reservoir at thermal equilibrium to derive this distribution. I under... | https://physics.stackexchange.com/questions/83325/logical-understanding-of-the-canonical-probability-distribution-canonical-ensem |
Question: <p>I seem to find a contradiction in the notion of probability density used by Landau and the notion of micro-canonical ensemble.</p>
<p>To see this, take an isolated classical system and we know experimentally that its energy lies between $E-\Delta$ and $E+\Delta$. So, we take a hypershell corresponding to ... | https://physics.stackexchange.com/questions/37740/micro-canonical-ensemble-and-classical-reality |
Question: <p>I read in a book on quantum fluctuations and quantum noise that, at thermal equilibrium the classical canonical variables are uncorrelated, ie: $$\langle xp\rangle=\langle x\rangle\langle p\rangle$$
But I am not sure to understand the sense of <em>at thermal equilibrium</em>, for me it just means $$\langle... | https://physics.stackexchange.com/questions/47709/thermal-equilibrium-and-non-correlations |
Question: <p>I need to consider a couple of examples of systems which have energies that are intensive variables - not extensive. I'be been thinking about this and I am not coming up with anything. My understanding is that extensive variables (at least wrt usual energies) scales with mass or length (system size). It al... | https://physics.stackexchange.com/questions/49789/examples-of-systems-with-energy-as-an-intensive-variable |
Question: <p>One can define entropy as $$S=k\log{\omega(E)},$$ where $\omega(E)$ is the numbers of states with energy equal $E$; and the canonical partition function for a set of N particles is defined as$$Z_N=\sum_{\phi}e^{-\beta E[\phi]}=e^{-\beta F(\beta,N)},$$ where the sum run on states $\phi$ and the free energy ... | https://physics.stackexchange.com/questions/53381/statistical-mechanic |
Question: <p>We know that in an isolated system, the density matrix is the microcanonical distribution matrix. That this the possibility for all the states with energy in a certain interval is a constant?
But how can I deduce this from the postulate of equal probability? </p>
Answer: <p>The assertion that the density ... | https://physics.stackexchange.com/questions/58102/microcanonical-distribution |
Question: <p>To derive Bose-Einstein and Fermi-Dirac distribution, we need to apply grand canonical ensemble:$Z(z,V,T)=\displaystyle\sum_{N=0}^{\infty}[z^N\sideset{}{'}\sum\limits_{\{n_j\}}e^{-\beta\sum\limits_{j}n_j\epsilon_j}]$. There is a constraint $\sideset{}{'}\sum\limits_{\{n_j\}}$ for quantum particles(bosons a... | https://physics.stackexchange.com/questions/60256/classical-quantum-particles-in-grand-canonical-ensemble |
Question: <p>Is it always assumed that, in a microcanonical ensemble, the number of particles is $N \gg 1$ ?</p>
<p>If no, are all the theorems related to the microcanonical description true even if the number of particles is small ?</p>
Answer: <p>Numerical simulations are a good example to see with your own eyes th... | https://physics.stackexchange.com/questions/62226/number-of-particles-in-a-microcanonical-ensemble |
Question: <p>Here's a simple 'derivation' of the Brownian motion law that after N steps of unit distance 1, the total distance from the origin will be sqrt(N) on average. It's certainly not rigorous, but I'm wondering if people think it's reasonable, or possibly even a commonly known.</p>
<ol>
<li><p>An object takes o... | https://physics.stackexchange.com/questions/12297/is-this-geometrical-derivation-of-brownian-motion-legitimate |
Question: <p>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?
<span class="math-container">$$
\chi = \frac{\partial M}{\partial H}
= \frac{1}{N}\sum_{i,j} \Gamma(i,j)
$$</span></p>
Answer: <p>I'll restrict my answer t... | https://physics.stackexchange.com/questions/434199/how-can-i-derive-the-analog-of-the-susceptibility-sum-rule-for-the-specific-heat |
Question: <p>The chemical potential of a noninteracting Bose gas can never be negative while that of a noninteracting Fermi gas can be both positive or negative. What can be said about the chemical potential of noninteracting classical ideal gas obeying MB distribution? </p>
Answer: <p>The simplest way to compute this... | https://physics.stackexchange.com/questions/454415/what-is-the-sign-of-chemical-potential-of-a-noninteracting-classical-ideal-gas-o |
Question: <p>I'm following Wikipedia's derivation of <a href="https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_statistics#Derivation_from_microcanonical_ensemble" rel="noreferrer">Maxwell-Boltzmann statistics</a>.</p>
<p>After applying Lagrange multipliers, we arrive at this expression for energy:</p>
<p><span... | https://physics.stackexchange.com/questions/461145/lagrange-multipliers-in-maxwell-boltzmann-statistics |
Question: <p>Is it possible for an electron to reside in an energy level lower than that of the ground state? What happens to the electrons when an atom is brought down to 0K , do they come closer? What happens to the left of the orbitals ? </p>
Answer: <p>As pointed out in a comment by another user in a previous post... | https://physics.stackexchange.com/questions/483835/electrons-residing-in-an-orbit-with-energy-lower-than-the-ground-state-energy |
Question: <p>Most of the cases when I see applications of statistical mechanics is when Fermi-Dirac or Bose-Einstein statistic are used in condensed matter or the equilibrium equation of neutron stars.</p>
<p>Besides the Poisson-Boltzmann equation for colloids and plasma screening, I would like to know what are the mo... | https://physics.stackexchange.com/questions/484734/modern-uses-of-classical-statistical-mechanics |
Question: <p>So, here is a system having two subsystems <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> where the two subsystems can exchange energy between them, then the total number of accessible microstates of the whole system is given by, <span class="math-container">$$... | https://physics.stackexchange.com/questions/547933/approximation-of-the-total-number-of-accessible-microstates |
Question: <p>I'm reading this article called:An experiment to demonstrate the canonical distribution(by M. D. Sturge and Song Bac Toha)
Department of Physics, Dartmouth College, Hanover, New Hampshire 03755.</p>
<p>They talked about the probability of a particle overcoming and energy barrier of height <span class="math... | https://physics.stackexchange.com/questions/619723/probability-of-particle-overcoming-an-energy-barrier |
Question: <ol>
<li><strong>Consider the three definition of entropy namely
<span class="math-container">\begin{eqnarray}
S &\equiv& k\log\Gamma(E), \label{1.1}\\
S &\equiv& k\log\Sigma(E), \label{1.2}\\
S &\equiv& k\log\omega(E), \label{1.3}
\end{eqnarray}</span>
where <span class="math-... | https://physics.stackexchange.com/questions/645667/show-that-these-definitions-are-equivalent |
Question: <p>For those wondering precisely what I am referencing throughout, I am contrasting the discussion in Reif Chapter 3.4 (general thermal equilibrium) with Chapter 6.2 (microcanonical ensemble).</p>
<p>In the most general approach to thermal equilibrium (let us consider subsystems A and A' in which no external ... | https://physics.stackexchange.com/questions/711617/contrasting-the-microcanonical-ensemble-with-the-general-approach-to-thermal-equ |
Question: <p>This is quoted from Daniel Schroeder's <em>An introduction to thermal Physics</em>:</p>
<blockquote>
<p><span class="math-container">$$\Omega= \left(\frac{e}{N}\right)^{2N} \; e^{N\ln (q/2)^2} e^{-N(2x/q)^2}\;=\; \Omega_\text{max} \cdot e^{-N(2x/q)^2}\;. $$</span></p>
<p>A function of this form is called <... | https://physics.stackexchange.com/questions/222543/sharpness-of-multiplicity-function |
Question: <p>I've been following Reif's <em>Fundamentals of Statistical and Thermal Physics</em>; there I came before the derivation of <a href="https://en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian)" rel="nofollow noreferrer">Liouville's theorem</a>:</p>
<p><a href="https://i.sstatic.net/puKHUl.png" rel="n... | https://physics.stackexchange.com/questions/237826/how-does-rho-dotq-1-mathrm-dt-mathrm-dq-2-ldots-mathrm-dp-f-represe |
Question: <p>Consider a mass m fixed to the middle point of a string of length $L$ whose extremities are a distance $$l$$ apart, and pulled with a tension $$F$$. The system is in thermal equilibrium, and one supposes that the only effect of thermal fluctuations is to make the system rotate about the horizontal (dashed)... | https://physics.stackexchange.com/questions/241561/average-value-of-force-in-rotating-bead-using-statistical-physics |
Question: <p>I mean that, is there anything more fundamental to yields the result that the random force in Langevin Equation is delta-correlated?<br/>
As is shown in the picture of a textbook below, its formula (3.4) is given by the assumption that "impacts are independent".However, it is still daunted for me to derive... | https://physics.stackexchange.com/questions/271440/how-to-interprete-that-the-random-forces-in-langevin-equation-are-assumed-to-be |
Question: <p>Recently I found <a href="https://ocw.mit.edu/courses/physics/8-044-statistical-physics-i-spring-2013/readings-notes-slides/MIT8_044S13_L1.pdf" rel="nofollow noreferrer">a very interesting map</a> which seems to contain all these elements one will meet in the statistical mechanics.</p>
<p>So does anybody w... | https://physics.stackexchange.com/questions/309878/how-to-understand-the-map-for-the-statistical-mechanics |
Question: <p>In Statistical thermodynamics Maxwell-Boltzmann statistics is considered a pre-quantum statistics. However in the mathematical treatment in all textbooks, and also in <a href="https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_statistics" rel="nofollow noreferrer">Wikipedia article</a>, there is the c... | https://physics.stackexchange.com/questions/587942/what-is-the-concept-of-energy-level-in-maxwell-boltzmann-statistics |
Question: <p>Suppose you have a container of volume V containing some gas with energy E and N particles. Let's assume the container to be isolated for now.</p>
<p>The microcanonical ensemble tells us that all microstates are equally likely. So a specific state in which all the molecules are at the top is as likely as ... | https://physics.stackexchange.com/questions/592443/confusion-about-the-second-law-of-thermodynamics-and-statistical-mechanics |
Question: <p>I'm trying to solve a problem regarding two systems interacting through a coupling hamiltonian:
<span class="math-container">$$ H(x,y) = H_1(x) + H_2(y) + kxy $$</span>
I am supposed to express the mean values <span class="math-container">$\langle x \rangle$</span> and <span class="math-container">$\langle... | https://physics.stackexchange.com/questions/593611/mean-values-of-position-in-van-der-waals-coupled-systems |
Question: <p>We have read the Fundamental postulate of statistical mechanics which says that :</p>
<blockquote>
<p>In a state of thermal equilibrium, All the accessible microstates of the system are equally probable.</p>
</blockquote>
<p>Suppose a system in thermal equilibrium with total energy to be <span class="math-... | https://physics.stackexchange.com/questions/596165/is-it-possible-for-a-particle-to-have-all-the-energy-of-the-isolated-system-of-p |
Question: <p>In Feynman's treatment of <a href="https://www.feynmanlectures.caltech.edu/I_40.html" rel="nofollow noreferrer">statistical mechanics</a>, <em>40–4 The distribution of molecular speeds</em>, Feynman found that the number of gas molecules per unit area per second who have an upper velocity component <span c... | https://physics.stackexchange.com/questions/522369/apparent-contradiction-in-feynmans-treatment-of-the-velocity-distribution-of-a |
Question: <p>Why when <span class="math-container">$-(\frac{\partial p}{\partial V})_T\geq 0$</span> we can say <span class="math-container">$-\frac{1}{V}(\frac{\partial V}{\partial p})_T\geq 0$</span> where V is the volume, p is the mean pressure of the system under consideration and T is the temperature which is kept... | https://physics.stackexchange.com/questions/535936/why-when-frac-partial-p-partial-v-t-geq-0-we-can-say-frac1v-fr |
Question: <p>I am working through Franz Schwabl's book on Statistical Mechanics, and he has a number of derivations of thermodynamic quantities that are different than those I have seen before. I am also having difficulty finding them repeated elsewhere.</p>
<p>In particular, he has a method for calculating <span class... | https://physics.stackexchange.com/questions/101406/qho-in-microcanonical-ensemble-problem-with-alternate-derivation |
Question: <p>I am writing code to find the heat capacity <span class="math-container">$C_v$</span> of a canonical <span class="math-container">$NVT$</span> ensemble. We know that,
<span class="math-container">$$C_v = \frac{\langle U^2 \rangle - \langle U \rangle ^2}{k_B T^2}$$</span></p>
<p>I have written a Metropolis ... | https://physics.stackexchange.com/questions/565081/calculating-c-v-of-a-canonical-ensemble |
Question: <p>The energy of a particle is given by E=|p|+|q|, where p and q are generalized momentum and coordinate respectively. All the states with E less than equal to E0 are equally probable and states with E greater than equal to E0 are inaccessible. What is the probability density of finding the particle at coordi... | https://physics.stackexchange.com/questions/244657/microcanonical-ensemle |
Question: <p>I am trying to mathematically show that the BBGKY hierarchy for s particles is time symmetric by setting $t\rightarrow -t$. Using the Wikipedia notation for the s-particle we have</p>
<p>$\frac{\partial f_s}{\partial t} + \sum_{i=1}^s \dot{\mathbf{q}}_i \frac{\partial f_s}{\partial \mathbf{q}_i} + \sum_{i... | https://physics.stackexchange.com/questions/249676/how-can-we-show-that-the-bbgky-hierarchy-is-time-symmetric |
Question: <p>Suppose that particles of two different species, A and B, can be chosen with
probability $p_A$ and $p_B$, respectively. </p>
<p>What would be the probability (and distribution) $p(N_A;N)$ that $N_A$ out of $N$ particles are of type A? </p>
<p>I'm trying to apply the Binomial distribution here but am both... | https://physics.stackexchange.com/questions/108566/probability-distribution-of-two-particle-types-system |
Question: <p>Three kinds of distributions.
The <strong>states occupation rates</strong>:</p>
<h2><strong>F.D.</strong> $n_{i}=\frac{1}{e^{\beta (\varepsilon _{i}-\mu )}+1}$ <strong>B.E.</strong> $n_{i}=\frac{1}{e^{\beta (\varepsilon _{i}-\mu )}-1}$ <strong>Boltzmann</strong> $n_{i}=e^{-\beta (\varepsilon _{i}-\mu )}$<... | https://physics.stackexchange.com/questions/114718/state-occupation-rate-n-i-frac1e-beta-varepsilon-i-mu-1-1-0 |
Question: <p>In the derivation of maxwell-boltzmann distributions, the method of Lagrange multiplier is </p>
<p>$\sum n_i = N$</p>
<p>$\sum n_i E_i = E$</p>
<p>where $N$ is the total number of particles, and $E$ is the total energy. And we try to find the macrostate with the most microstates, I think the derivation ... | https://physics.stackexchange.com/questions/137433/ensembles-and-lagrange-multipliers |
Question: <p>If I choose the number of microstates for energy $E$ to be $\Omega(E) = e^{aE}$ ($a>0$), its temperature is constant:
$$
kT = \left( {d\ln \Omega \over dE} \right)^{-1} = 1/a
$$
If I choose $\Omega(E) = e^{aE^2}$ ($a>0$), its temperature decreases as the energy goes up:
$$
kT = \left( {d\ln \Omega \o... | https://physics.stackexchange.com/questions/137501/exponentially-increasing-omegae |
Question: <p>This is related with Equation 8.58 in Kerson Huang's 2nd edition of Statistical Mechanics. </p>
<p>The partition functions for the ideal gases are given as
$ Q_N (V,T) =\sum_{\{ n_p \}} g\{n_p \}e^{-\beta E\{n_p \}} $ where $E\{n_p \} =\sum_p \epsilon_p n_p$ and the occupation numbers are subject to the $... | https://physics.stackexchange.com/questions/142723/the-number-of-states-for-fermions-bosons-and-boltzman-in-statistical-mechanics |
Question: <p>What does Born Green equation obtained from YBG hierarchy for the equilibrium particle densities signify? I mean how can you model the equation into a physical problem?I understood the steps involved in the derivation of the expression, but am still unsure if I understood it physically, am unable to explai... | https://physics.stackexchange.com/questions/149561/what-does-born-green-equation-signify-physically |
Question: <p>What is a "microscopic realization" of a system?</p>
<p>The context is statistical mechanics. The microscopic system consists of many atoms (too many to track individually) with an assigned probability density function <code>f(x,y,z,Vx,Vy,Vz,t)</code>. </p>
<p>The macroscopic system consists of the atoms... | https://physics.stackexchange.com/questions/133720/statistical-mechanics-what-is-a-microscopic-realization-of-a-system |
Question: <p>The collision term in the Boltzmann equation can be derived from the BBGKY hierarchy. </p>
<p><a href="http://en.wikipedia.org/wiki/BBGKY_hierarchy" rel="nofollow">Wikipedia</a> says:</p>
<blockquote>
<p>In statistical physics, the BBGKY hierarchy [...] is a set of equations describing the dynamics of ... | https://physics.stackexchange.com/questions/133787/the-bbgky-hierarchy |
Question: <p>Let's have Boltzmann collision integral:
$$
I_{coll} =\int d \sigma d^{3}\mathbf p_{1}(ff_{1} - f{'}f{'}_{1})|\mathbf v_{rel}|.\tag{1}\label{1}
$$
How to transform $\eqref{1}$ to BGK collision integral,
$$
I_{coll} = \frac{1}{\tau}(f - f_{0})?\tag{2}\label{2}
$$
Here $\tau = \frac{|v_{rel}|}{l} \sim v_{rel... | https://physics.stackexchange.com/questions/154371/how-to-derive-the-bhatnagar-gross-krook-collision-integral-from-boltzmann-one |
Question: <p>I am trying working on a problem in which there are two energy states $E_{1}<E_{2}$, and three different (i.e. distinguishable) particles. </p>
<p>I cannot decide if the order of the particles matters. If it doesn't, then there are 8 states. If order does matter, there are 24. My problem is not knowing... | https://physics.stackexchange.com/questions/166369/number-of-states-of-a-simple-system |
Question: <p>MB distribution is followed if there are N no. of non interacting and distinguishable particles. But if N tends to infinity why does the no. of micro states reduces?
Is there any peak in the graph?</p>
Answer: | https://physics.stackexchange.com/questions/171499/how-will-the-distribution-of-the-no-of-particles-be-in-a-system-n-v-e-if-n-t |
Question: <p>Thermodynamics deals with "equilibrium states of macroscopic matter", that is, considering macroscopic systems there are states which can be characterized fully by a few number of measured degrees of freedom and on such states we are not able, through macroscopic measurements to see the fact that the molec... | https://physics.stackexchange.com/questions/174627/statistical-mechanics-deals-with-the-same-systems-that-thermodynamics-does |
Question: <p>I am reading "A Modern Course in Statistical Mechanics" by Linda E. Reichl. Where i encountered this notation:</p>
<p>$$\Delta S = \bar g : \vec \alpha \vec \alpha$$</p>
<p>Here $\bar g$ is $$ g_{i,j}=-{ \partial^2 S \over \partial A_i \partial A_j}\bigg|_{A_i = A_i^0, A_j = A_j^0 } $$
a matrix of seco... | https://physics.stackexchange.com/questions/182773/strange-vector-matrix-operation-in-a-modern-course-in-statistical-physics-by |
Question: <p>Do bending moment and shear force of a beam depend on it's cross sectional dimentions??
Since all the diagrams which I have draw so far don't involve any cross section details. So I think they do not depend on them and don't do any influences on the shear force diagrams and the bending moment diagram.</p>
... | https://physics.stackexchange.com/questions/187147/bending-moment-and-shear-force |
Question: <p>It's a common fact that in physics, we use statistics (or maybe probabilities ) to describe the behaviour of a system. It was from the statistical analysis of a system where quantum statistics arose and then the theory of quantum mechanics began.</p>
<p>How is possible to make such descriptions and to con... | https://physics.stackexchange.com/questions/187970/how-statistical-physics |
Question: <p>Consider a system whose sole degree of freedom is an angle $\theta$ that goes from $0$ to $2\pi$. Let $E(\theta)$ be its energy function. Obviously, $E(\theta)$ is $2\pi$-periodic. What's the general form for the Boltzmann distribution for $\theta$? Is it just: $P(\theta)\propto e^{-E(\theta)/kT}$? Or is t... | https://physics.stackexchange.com/questions/189522/boltzmann-distribution-for-angles |
Question: <p>In wear test of pin on disc apparatus i found that mass loss of pin when i used Aluminum disc is higher than when i used Steel disc under the same conditions ,pressure, velocity and contact time can anyone explain this behavior to me and give me the reason ?</p>
Answer: <p>Sometimes, when you have a soft ... | https://physics.stackexchange.com/questions/203401/the-different-in-wear-test-when-using-aluminum-and-steel-disc-in-pin-on-disc-app |
Question: <p>Suppose a system A which is a vessel of water with two electrodes, connected by a resistor, placed in the water. </p>
<p>If you apply voltage to the electrodes, energy is transferred from the battery (not included in system A) to system A.</p>
<p>I read in a book that the form of energy transferred is wo... | https://physics.stackexchange.com/questions/205394/energy-transfer-in-form-of-work-or-heat |
Question: <p>Why we consider the maximum number of micro states or complexions as equilibrium state of a macro state or a system in statistical physics?</p>
Answer: <p>In statistical mechanics all micro-states are considered to be equally likely. This means that the most likely macro-state is the one that contains the... | https://physics.stackexchange.com/questions/211202/equilibrium-state-of-a-system-in-statistical-mechanics |
Question: <p>When I read Books about statistical physics, then often names like "macroscopic variable / observable", parameter of the macroscopic state and generalized force are used, and I want to know, what is the difference, and wether there are definitions for that. Plus, I want to know of what type are the commonl... | https://physics.stackexchange.com/questions/228668/difference-between-macroscopic-variable-macroscopic-observable-parameter-and-g |
Question: <p>How is
$$ \left(1-\frac{p^2}{2mE}\right)^{3N/2-2} = \exp\left(-\frac{3N}{2}\frac{p^2}{2mE}\right)$$
(Karder, Statistical Physics of Particles, Page 107)</p>
<p>in the large $E$ limit. Here $N$ is particle, of the order of $10^{23}$, $E$ is the total energy.
I roughly guess that it should be $\exp(-\frac{... | https://physics.stackexchange.com/questions/227654/how-is-left1-fracp22me-right3n-2-2-exp-left-frac3n2-fra |
Question: <p>Can we interpret the <strong>Euler product formula</strong> " $\sum\frac{1}{n^s} = \prod_{p\;\mathrm{prime}} \frac{1}{1-p^{-s}} $ " in a stat. physical sense, as a product of single-particle system <em>partition functions</em>, considering them <em>statistically independent</em> ?</p>
Answer: <p>Umm...OK ... | https://physics.stackexchange.com/questions/229111/partition-function-of-primon-bosonic-gas |
Question: <p>The Fundamental Postulate says:</p>
<blockquote>
<p>In <em>equilibrium</em>, all accessible microstates are equally likely.</p>
</blockquote>
<p>Accessible means having same energy.(right?)</p>
<p>Let a container is taken full of gas having number of particles $N_,$ volume $V$ and energy $E\:_;$ the s... | https://physics.stackexchange.com/questions/234396/does-fluctuation-really-occur-in-equilibrium-as-its-microstates-are-allowed-to-o |
Question: <p>Knowing that, in the $\mu$-canonical (or micro-canonical) and canonical ensembles, the number of particles is held constant and usually reflects the actual number of particles, in which case $N_{MC} \simeq N_{actual}$, and also $N_C \simeq N_{actual}$. Or otherwise </p>
<p>$\text{sgn}(N_{MC}) = \text{sgn}... | https://physics.stackexchange.com/questions/210152/negative-amount-of-particles-in-a-grand-canonical-ensemble |
Question: <p>How can I derive the Fermi-Dirac distribution function using simple mathematics?
I am now tired of looking for the derivation on the net.So please help me to understand how actually electrons are distributed between various energy levels?</p>
Answer: <p>Okay, so do you understand derivation of thermodynam... | https://physics.stackexchange.com/questions/240623/derivation-of-fermi-dirac-distribution |
Question: <p>The energy for dipoles in a magnetic field can be described by <span class="math-container">$$H = - \mathbf{m} \cdot \mathbf{B}.$$</span> What I did for an exercise was integrate the partition function for the case where I allow rotation in three dimensions (reducing it to the integral <span class="math-co... | https://physics.stackexchange.com/questions/677796/langevin-paramagnetism-for-dipoles-rotating-in-2d |
Question: <p>In lecture we introduced the average energy for a single particle as <span class="math-container">$$E = \langle \varepsilon \rangle = \frac{\int d \varepsilon g(\varepsilon) e^{- \beta \varepsilon} \varepsilon}{Z_1}$$</span>
where <span class="math-container">$Z_1 = \int d \varepsilon e^{- \beta \varepsilo... | https://physics.stackexchange.com/questions/679293/question-regarding-expectation-value-of-energy-and-gibbs-factor |
Question: <p>I am trying to find the nr. of microstates inside a box of dimensions <span class="math-container">$L_1,L_2,L_3$</span> of a hypersphere in the phase space.We have a gas of N particles in 3D. While we have a 6N dimensional phase space because we assumed that the gas is ideal, we do not have potential energ... | https://physics.stackexchange.com/questions/682081/mce-nr-of-microstates-proof-that-entropy-is-extensive-ideal-gas-in-a-box |
Question: <p>In statistical mechanics, is the partition function a mathematical cumulative distribution function as in probability mathematics ?</p>
<p>Why is the partition function the fundamental objects for statistical mechanisms ?</p>
<p>Why isn't it the probability <em>density</em> function ?</p>
Answer: | https://physics.stackexchange.com/questions/685153/in-statistical-mechanics-is-the-partition-function-a-mathematical-cumulative-di |
Question: <p>There are two boxes (which I will call 1 and 2) that are initially thermally isolated and have a sliding door in between them. We can write the probability of configuration <span class="math-container">$A$</span> in box 1 as,</p>
<p><span class="math-container">$$P_1(A)=\frac{1}{\Omega_1}$$</span></p>
<p>S... | https://physics.stackexchange.com/questions/693650/microstates-and-combinations |
Question: <p>The Helmholtz-free energy from classical thermo is defined as
$$\text{F=u-TS}$$</p>
<p>taking the differential and algebraic manipulation, we arrive at</p>
<p>$$\text{dF=-pdv-sdT}$$</p>
<p>Observe that:</p>
<p>$$\text{p=-(}\frac{\text{$\delta $F}\backslash }{\text{$\delta $v}})_T$$ and $$\text{S=-(}\f... | https://physics.stackexchange.com/questions/182774/bridging-the-connection-from-the-helmholtz-free-energy-in-classical-thermo-to-st |
Question: <p>If we consider the average occupancy for a bose gas, we know:</p>
<p><span class="math-container">$$\langle n \rangle_B=\frac{1}{e^{\beta(\epsilon-\mu)} -1}$$</span></p>
<p>I also know that for high temperatures this transforms itself to the occupancy for a classical Boltzmann gas:</p>
<p><span class="math... | https://physics.stackexchange.com/questions/694716/occupation-number-of-a-bosonic-gas-for-high-temperatures |
Question: <p>Susceptibility can be espressed in terms of Gibbs free energy as:</p>
<p><span class="math-container">$$\chi^{-1}= \frac{\partial^2g}{\partial m^2}$$</span></p>
<p>Where <span class="math-container">$g$</span> is the intensive Gibbs free energy. So if the second derivative of <span class="math-container">$... | https://physics.stackexchange.com/questions/694727/can-susceptibility-be-infinite-even-for-small-systems |
Question: <p>I'm trying to understand the Maxwell-Boltzman Distribution, and in particular the derivation from the boltzman distribution for energy. I have successfully created an incorrect derivation, but I'm not sure what's wrong with it :). Any guidance would be much appreciated!</p>
<p>I believe that the probabili... | https://physics.stackexchange.com/questions/106288/boltzmann-distribution-derivation-from-canonical-distribution |
Question: <p>I've just begun learning Statistical Mechanics and my question concerns my professor's statement and I quote:</p>
<blockquote>
<p>Consider a gas of <span class="math-container">$N$</span> atoms which can have position <span class="math-container">$q_i$</span> and momentum <span class="math-container">$p_i$... | https://physics.stackexchange.com/questions/695242/phase-space-diagrams-of-microstates-infinite-dimensional |
Question: <p>In my textbook, the definition of temperature begins by determining the maximum number of microstates for two systems in thermal equilibrium of energies <span class="math-container">$E_1$</span> and <span class="math-container">$E_2$</span> and microstates <span class="math-container">$\Omega_1(E_1)$</span... | https://physics.stackexchange.com/questions/705610/how-do-we-know-there-are-no-local-maxima-in-the-number-of-microstates-with-respe |
Question: <p>Consider that the position and momentum of my particles have a Gaussian distribution.
If I now calculate the number of particles in
<span class="math-container">$dx$</span> with momentum <span class="math-container">$dp$</span> then which one of the following would it be:</p>
<p><span class="math-container... | https://physics.stackexchange.com/questions/711853/what-is-the-particle-density-for-a-gaussian-distribution-in-position-and-velocit |
Question: <p>I am looking through derivations of the Boltzmann distribution. The method I've seen uses an argument that involves counting distinguishable microstates of a system with fixed energy, and then assuming that these distinguishable microstates are equally likely to occur.</p>
<p>A first assumption is that fro... | https://physics.stackexchange.com/questions/712644/boltzmann-distribution-why-does-distinguishability-increase-likelihood |
Question: <p>I am confused about the fundamental assumption of statistical mechanics. It says, over a long time scale, that all microstates are equally accessible. I get it so far.</p>
<p>But for microstate, there are arrangements such that no particles(thinking of Einstein solid) have energy (Daniel schroeder), how is... | https://physics.stackexchange.com/questions/713939/confusion-about-fundamental-assumption-of-statistical-mechanics |
Question: <p>I've tried to derive the pressure, <span class="math-container">$P$</span>, for a weakly degenerate gas of fermions (and analogously for bosons). The strange thing is that the expression I calculated is correct except for the sign of a term. I checked the calculations and i don't see any error, so the erro... | https://physics.stackexchange.com/questions/729892/fermions-and-bosons-weakly-degenerate-gases |
Question: <p><a href="http://en.wikipedia.org/wiki/Potts_model">Wikipedia</a> writes:</p>
<blockquote>
<p>In statistical mechanics, the Potts
model, a generalization of the Ising
model, is a model of interacting spins
on a crystalline lattice.</p>
</blockquote>
<p>From combinatorics conferences and seminars, ... | https://physics.stackexchange.com/questions/482/what-does-the-chromatic-polynomial-have-to-do-with-the-potts-model |
Question: <p>In a interaction-round-a-face model of $n^2$ particles in a lattice, a weight $W(a,b,c,d)$ is assigned to each face in the lattice based on the spins $a,b,c,d$ (listed say from the bottom-left corner in counter-clockwise fashion) of the particles at the corners of the face. Based on this, a partition func... | https://physics.stackexchange.com/questions/6899/applicability-of-baxters-method-for-irf-models |
Question: <p>There are many articles published in physics journals about <a href="http://en.wikipedia.org/wiki/Flocking_%28behavior%29" rel="nofollow">flocking</a>. Is there a physical reason for these phenomena or is it just because physics methods are being used to study collective motion?</p>
<p>It seems there is n... | https://physics.stackexchange.com/questions/10578/is-there-any-physics-behind-flocking |
Question: <p>I really need some very simple discussions of random walk (probability). Couldn't get anything from class, more so from Reif. Thanks!</p>
Answer: <p>Random walk is intimately connected with diffusion, heat equation, Laplacian, harmonic functions, quadratic forms and Gaussian distribution. Here's a sketch ... | https://physics.stackexchange.com/questions/12733/can-somebody-provide-some-sort-of-crash-course-on-random-walk-and-its-problems-a |
Question: <p>I'm reading <em>An Introduction to Stochastic Processes in Physics</em> by Don S Lemons. Problem 10.2 leads to a pair of equations:</p>
<p>$dV_x = -\gamma V_xdt+V_y\Omega dt-V_y\sqrt{2\gamma dt}N_t(0,1)$</p>
<p>$dV_y = -\gamma V_ydt-V_x\Omega dt+V_x\sqrt{2\gamma dt}N_t(0,1)$</p>
<p>($\gamma$, $\Omega$ ... | https://physics.stackexchange.com/questions/13804/cross-field-diffusion-from-smoluchowski-approximation |
Question: <p>When I learned anharmonic model of crystal, I read that considering anharmonic oscillations and Boltzmann distribution for the "atoms" of crystal we can get the dependence of distance between the "atoms" from a temperature as</p>
<p>$$
\langle r \rangle = r_{0} + \alpha T.
$$</p>
<p>As I understood th... | https://physics.stackexchange.com/questions/28529/how-to-take-this-integral |
Question: <p>I am trying to solve a problem from the book 'Introductory Statistical Mechanics' (Bowley, Sanchez). The question reads:</p>
<p>Calculate the free energy of a system of N particles, each with spin 3/2 with one particle per site, given that the levels associated with the four spin states have energies e, 2... | https://physics.stackexchange.com/questions/43625/spin-3-2-statistical-mechanics-problem |
Question: <p>Why do we study these different ensembles, microcanonical, canonical, grand canonical ensemble ? Are they used for studying different physical system or scenarios?(e.g. in some system you can only treat it as mirocanonical, and in other cases you can only apply canonical ensemble) Do they have the same res... | https://physics.stackexchange.com/questions/54758/why-do-we-need-different-ensembles-in-statistical-mechanics |
Question: <p>Let $D_{ij}$ a random matrix with i.i.d positive coefficients. One can take for instance $D_{ij}$ uniformly distributed in [0,1]. We consider the following energy function $H(x)$ defined for $x=(x_i)_1^n$, with each $x_i\in \mathbb{R}^k$, where $n>k$ are two positive integers:</p>
<p>$$H(x) = \sum_{i,j... | https://physics.stackexchange.com/questions/55525/partition-function-for-multidimensional-scaling-energy |
Question: <p>In Bose-Einstein condensation, the chemical potential is less than the ground state energy of the system($\mu<\epsilon_g$). But why does the massless boson such as photon have zero chemichal potential($\mu=0$)?</p>
Answer: <p>The chemical potential is a complementary variable to $N$, the number of part... | https://physics.stackexchange.com/questions/60499/why-the-chemical-potential-of-massless-boson-is-zero |
Question: <p>I know that <a href="http://en.wikipedia.org/wiki/Classical_XY_model" rel="nofollow">XY statistical model</a> for $d=2$ doesn't show a regular phase transition , while the $3d$ has, I was wondering what is the behaviour for $2< d < 3$.</p>
<p>If it is simpler one could consider another model in its ... | https://physics.stackexchange.com/questions/64552/what-is-the-minimum-non-integer-dimension-for-which-the-xy-model-shows-a-phase-t |
Question: <p>If one looks at a cubic box of gaseous atoms all initially flying in the same direction at the same speed (but flying at an angle to the walls, so as not to reflect up-and-down against the box walls forever), they will collide with the walls and each other, their previously uniform velocities becoming mess... | https://physics.stackexchange.com/questions/74660/why-is-velocity-normally-distributed-in-a-gas-but-not-energy |
Question: <p>I'm currently doing some calculations which require evaluating various standard thermal expectation values in the canonical ensemble (both bosons and fermions). Now, in order to make my theoretical machinations easier, I am actually using the grand canonical ensemble, where the chemical potential acts as a... | https://physics.stackexchange.com/questions/80050/practical-difference-between-canonical-and-grand-canonical-ensembles |
Question: <p>Looking at <a href="http://www.tcm.phy.cam.ac.uk/~bds10/phase/scaling.pdf" rel="nofollow">this paper on page 1</a> how is the first limit obtained? That is, if I have some homogeneous function $g_f(h/t^{\Delta})$, how does setting the gap exponent $\Delta$ to $3/2$ ensure that $$\lim_{x \to 0} g_f(x) = -1/... | https://physics.stackexchange.com/questions/89637/gap-exponents-and-homogeneous-functions |
Question: <p>I'm studying different ensembles and different statistics (<a href="http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution" rel="nofollow">M-B</a>, <a href="http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics" rel="nofollow">B-E</a>, <a href="http://en.wikipedia.org/wiki/Fermi%E2%80%... | https://physics.stackexchange.com/questions/89850/what-are-the-key-properties-of-and-differences-between-classical-and-quantum-sta |
Question: <p>I understand that <a href="http://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution" rel="nofollow">Maxwell-Boltzmann distributions</a> arise for distributions of weakly interacting particles at equilibrium. But I'd like to know if there's a deeper reason behind why they are specifically Maxwell... | https://physics.stackexchange.com/questions/91708/where-does-the-maxwell-boltzmann-distribution-come-from |
Question: <p>Consider an ensemble of $N\to\infty$ free particles, each of which can assume energy states $E_i\in\{0,E\}$. Using the canonical ensemble one can compute the occupation probability for a single of those particles to be in the excited state $E_i=E$ (or equivalently the expectation value for what fraction of... | https://physics.stackexchange.com/questions/93075/fermi-dirac-like-occupation-probability-at-high-temperature |
Question: <p>I'm studying statistical mechanics, in particular classical regime for Fermi Dirac and Bose Einstein gases. Time average value for occupation numbers in FDBE statistics:
$$ \langle n_\epsilon\rangle_{FB} = \frac{1}{e^{(\epsilon-\mu)\beta}\pm1} $$
For Boltzmann Statistics:
$$ \langle n_\epsilon \rangle_B = ... | https://physics.stackexchange.com/questions/93638/classical-regime-for-fermi-dirac-and-bose-einstein-gases |
Question: <p>I'm interested in the relation between the probability distribution $p_i$ over states of a system on the one side and the density of states $\rho(\eta)$ of its environment. (Meaning, $\int_{\eta_a}^{\eta_b} \rho(\eta) ~ \mathrm{d} \eta$ is the number of environment states with energies in the interval $[\e... | https://physics.stackexchange.com/questions/95174/boltzmann-gibbs-distribution-as-resulting-from-a-limiting-density-of-states |
Question: <p>I am trying to understand where the Boltzmann distribution comes from. I recently learned some interesting things of which my interpretation follows below. Did I interpret correctly? If so, is this all there is to it, or is this only part of the story?</p>
<p>When sampling coordinates in a high-dimensiona... | https://physics.stackexchange.com/questions/103453/the-maxwell-and-the-boltzmann-distributions |
Question: <p>I'd like to ask some questions about flipping two coins related to statistical mechanics, e.g. microcanonical distribution, phase space distribution function etc... after I rephrase the coin flipping problem into the language of statistical mechanics.</p>
<p>In probability theory, given the following prob... | https://physics.stackexchange.com/questions/108156/statistical-mechanics-of-a-coin-toss |
Question: <p>If you take the Maxwell-Boltzmann distribution of velocities (which depends on the mass) and substitute $v=\sqrt{\frac{2E}{m}}$ you get the distribution for the energies, which turns out to be independent of mass. What physical reality does this reflect? Why is the velocity distribution mass dependent, whe... | https://physics.stackexchange.com/questions/119739/what-is-an-intuitive-explantion-for-the-fact-that-the-maxwell-boltzmann-distribu |
Question: <p>In deriving the magnetic susceptibility of free electrons, we need to calculate</p>
<p>$$\chi = \left( \frac{\partial M}{\partial H} \right)_N = - \left( \frac{\partial^2 F}{\partial H^2} \right)_N.$$</p>
<p>Here, $F = E- TS $ is the free energy. It should be emphasized that the particle number $N$ is he... | https://physics.stackexchange.com/questions/122081/derivation-of-landau-diamagnetism |
Question: <p>What does "accessibility" mean in statistical mechanics?</p>
<p>Is it an equivalent concept to accessibility in mathematical control theory?</p>
<p>I'll provide an example:
When two systems A and B interact on a subspace of their respective spaces, i.e. where they overlap in space, does accessibility of... | https://physics.stackexchange.com/questions/133608/statistical-mechanics-meaning-of-accessible-in-accessible-microstates |
Question: <p>While doing a course in statistical physics I came across a term called coefficient of variation. Now according to <a href="http://en.wikipedia.org/wiki/Coefficient_of_variation" rel="nofollow">Wikipedia</a>, coefficient of variation</p>
<blockquote>
<p>shows the extent of variability in relation to mea... | https://physics.stackexchange.com/questions/139866/physical-meaning-of-coefficient-of-variation |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.