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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Partition Function for a Ideal Gas - Statistical Mechanics

While I was studying statistical mechanics, I saw this in the book that I'm following: We can divide the partition function into a product, $$ \zeta = \zeta_\text{trans}\zeta_\text{int} $$ where $\...
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Average speed of molecular effusion [closed]

I need know what is the average speed of molecular effusion I mean the speed of a particle that scape from a container. NOT the rate of molecular effusion.
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How do you determine the Einstein Temperature from Einsteins Heat capacity function? [duplicate]

I have a graph of Temperature K vs Heat Capacity C_v/k_B. The function it is fitted to is 3* x**2 * exp(x) / (exp(x) - 1)**2. Essentially i cant extract x from this equation. I am struggling to ...
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How do you determine the Einstein temperature from the Einstein heat capacity?

I have a graph of Temperature K vs Heat Capacity C_v/k_B. The function it is fitted to is 3* x**2 * exp(x) / (exp(x) - 1)**2. Essentially i cant extract x from this equation. I am struggling to ...
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37 views

Physical explanation of temperature dependence of chemical potential

I have recently started my first course on statistical mechanics and have been learning about the Fermion gas. I was calculating the temperature dependence of chemical potential for an electron gas ...
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Possibility/nature of SSB in systems with long-range interaction not mediated by gauge fields

Is there any real Condensed Matter system modelled by a Hamiltonian with long-range interaction (except those mediated by any gauge fields) such as long-range spin-spin interaction that displays the ...
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Problems in deriving second law of thermodynamics in terms of Gibbs energy

I want to express the second law of thermodynamics in terms of Gibbs free energy, $dG \leq 0$. It is true that: $$dH=TdS+Vdp+\sum_i\mu_idn_i$$ and $$dG=-SdT+Vdp+\sum_i\mu_idn_i$$ So for constant $T,...
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Help with 2nd Law and irreversibilty

This question is about the seemingly idealized notion of isolated systems and truly irreversible processes in the context of the 2nd Law. Here are the definitions and citations I'll use then my ...
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31 views

How can ergodicity explain thermalization

I am reading up on thermalization in classical systems. As most systems are ergodic, mostly through the mechanism of dynamical chaos, they will explore their whole allowed phasespace and we can ...
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1answer
44 views

Entropy as a state function - Is it just a postulate of the second principle?

I read quite a few questions on this website dealing with the idea of demonstrating that entropy is a state function. None of the answers I read seemed to be fully conclusive. So my question is : is ...
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How to unify the cumulant expansion and Feynman diagram expansion?

In most QFT books, the perturbation theory is given by "Taylor expansion". When evaluating 2-points, the numerator gives all the diagrams, i.e. $$\int D[\phi]e^{iS[\phi]}\phi_1\phi_2=\int D[\phi]\...
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Entropy generation at the molecular level in a irreversible process

When we expand an gas irreversibly in an adiabatic process then there is intermolecular friction, but what exactly gets transferred to heat. I have read that the directed motion gets randomized. But ...
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Expectation value in zero and nonzero temperature

Context: As a fundamental rule, in zero temperature, we know that the expectation value of a time-independent observable at time $t$, namely $\langle \hat{O}\rangle _t = \langle G(t)\mid \hat{O}\mid ...
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Can I derive the Maxwell-Boltzmann Statistics with rotational energy using this method?

I understand that the derivation of the Maxwell-Boltzmann statistics is based on particles only having translational kinetic energy and that it is derived by maximizing the number of ways $\Omega_T$ a ...
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2answers
42 views

A proof that internal non-conservative forces cannot change the total energy of an isolated system

In the absence of external work, and heat transfer, the total energy of a system (mechanical + thermal) will remain constant. Such a system is commonly referred to as isolated. Internal conservative ...
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Physical difference between Microcanonical and canonical ensemble

While performing a molecular dynamics simulation, what is physically different between holding the total energy constant and temperature constant? I have seen MD simulation studies where particle ...
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1answer
70 views

Proof for $\oint \frac{dQ}{T}=0 $ in a reversible process

I'm actually trying to prove that Entropy is a state function. I get struck at the point where I need to prove that $\oint \frac{dQ}{T}=0 $ for a reversible process. Clausius in his book The ...
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Formally define temperature as derivative of max gibbs entropy?

This question asks whether we can define the temperature in terms of the Gibbs entropy in the case of the canonical ensemble. In this question I want to ask whether we can define temperature in terms ...
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1answer
44 views

From Boltzmann equation to Lattice Boltzmann

I'm following the book Lattice Gas Cellular Automata and Lattice Boltzmann models which refers to this paper to explain how to discretize the Boltzmann equation (BE) into the Lattice Boltzmann ...
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Finding rate of change of momentum in a fire extinguisher [closed]

I am trying to find the rate of change of momentum as asked by part ii. I have begun the question by saying "dp/dt= -dN/dt x momentum of a particle". To cancel out the sqrt(pi) factor in dN/dt it ...
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3answers
45 views

Temperature dependence of entropy

$$\text{Entropy}=\frac{\text{Heat absorbed}}{Temperature}$$ $$\Rightarrow S=\frac{Q}{T}$$ $$[S]=[ML^2 T^{-2} K^{-1}]$$ If entropy increases with increase in temperature of the system, then it ...
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3answers
75 views

Entropy change in a reversible and Irreversible path

Let's consider 2 cases. First where a system is taken from state 1 to 2 in a reversible path. Second where the same system is taken from state 1 to 2 in an irreversible path. Can we say that Entropy ...
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1answer
77 views

What is the temperature of a pure quantum state?

I was wondering about temperatures and pure quantum states. I'm currently working on thermalization of isolated quantum systems, which can be described by pure quantum states (kets). How do we define ...
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How many degrees of freedom does the air have?

Very simple question that I am overthinking... But how many degrees of freedom does the air have? Assuming let's say the air is confined in a rigid box.
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2answers
187 views

Is entropy $S$ a fundamental quantity like Temperature?

What I'm trying to say is that $$S=\int\limits_{T_1}^{T_2}\frac{\mathrm dQ}{T}\tag{1}$$ depends only on the initial and final states. Why is that so? Is it like a "law" (like Newton's law of gravity ...
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Additional resources to Landau's approach to entropy

I extensively studied thermodynamics but was never really satisfied with how entropy is treated in this theory. Some time ago I started reading Landaus statistical physics. I find most of the book ...
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Is there a physics law according to which, in general, the smaller an object is, the faster it moves? [closed]

I have always felt, in general, that dogs run faster than humans and that birds fly faster than dogs and than bees can fly or at least drift by the wind faster than birds and that plant seeds would ...
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1answer
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Summation notation for statistical mechanics

I was taking a look to a book of statistical mechanics, many equations show something as follows: $$ Q(K,N) = \sum_{s_{1},s_{2},...,s_{N}=\pm 1}[ e^{K(...+s_1s_2+s_2s_3+s_3s_4...)} ]\tag{1}$$ then ...
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If liquid and gas are both chaotic states of matter, what's the difference between them on the molecular level?

I'm a laywoman in physics and recently found myself pondering about the matter reflected in the title of this post. To make my question more precise from the mathematical standpoint, let's suppose ...
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2answers
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Change of variables in an expression involving differentials

I'm trying to derive Maxwell-Boltzmann's distribution using the statistics of the classical canonical ensemble. Doing some operations, I have found out that the probability that a molecule of a gas ...
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2answers
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Why is the energy probability different from its speed probability in the Maxwell-Boltzmann Distribution?

I know this has been asked before here and I understand how the formula changes when the Maxwell-Boltzmann Distribution (MBD) is written in terms of speed. But I am trying to understand this more ...
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What is the fundamental mechanism(s) that allow energy to be stored in a compressed gas?

I thought I could simply Google this question and get a straight, simple answer. Everything but. Is it simply that the mean velocities of the molecules are increased (so molecular kinetic energy) as ...
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1answer
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About the $6N$ dimension when describing the Boltzmann equation

I have a question related with the formulation of the Boltzmann equation. In all the documents I read, it appears that the system is made of $N$ molecules, and so the phase space has $6N$ dimensions (...
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Time rate of thermodynamic potentials

In equilibrium thermodynamics we're concerned with the change in thermodynamic potentials between two equilibrium states (e.g. $\Delta U$, $\Delta G$, etc.), and we use these changes to make ...
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Why don't ensembles in statistical mechanics explicitly account for all conserved quantities? [duplicate]

Let's take the micro-canonical ensemble of an ideal gas. Essentially, for any given energy level $E$, it consists of a uniform distribution over micro-states that are consistent with energy level $E$, ...
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Can you add the density of states of two mono-atomic gases?

Say I have a system of 2 gases with $N_1$ and $N_2$ particles, each with respective masses $m_1$ and $m_2$. Would I be able to find the density of states for this system of two mixed gases $p(E)$ by ...
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1answer
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How is the Maxwell-Boltzmann Distribution a Chi-square Distribution?

This Wikipedia states that the MB Distribution in terms of energy is a Chi-square Distribution with 3 degrees of freedom. I know that the probability density formula of a Chi-square Distribution with ...
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Definition Chemical Potential

Chemical potential is a change in internal energy when a particle is added to system. now what if a system contain lots of energy state in this case if you add a particle to two distinct energy ...
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1answer
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Unconditional probability of for a system microstate in canonical ensemble

In Kardar's 'Statistical Physics of Particles', it is stated that the unconditional probability for a microstate $\mu_S$ of system $S$ (in a canonical ensemble made using a system $S$ and reservoir $R$...
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134 views

Taylor Series of a logarithmic function

I was reading Intro to Modern Statistical Mechanics by David Chandler, on page 63. He states the following: we can expand $\ln\Omega(E-E_v)$ in the Taylor series $$\ln\Omega(E-E_v) = \ln\Omega(E) - ...
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How can a system of 2 gases be in thermical equilibirum but not in mechanical or chemical equilibrium?

This question is asked on my Thermodynamics course, and I don't know the answer, at least to the 1st part. Regarding the 2nd part: I do believe that some chemical reactions may occur when 2 reactants ...
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Help on calculation in Zwanzig, “Nonequilibrium Statistical Mechanics”

I'm trying to verify a calculation from Zwanzig's Nonequilibrium Statistical Mechanics text from chapter two, deriving the Fokker-Planck Equation from a Langevin Equation, specifically equation (2.41)....
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Counting the number of microstates given the constraint

While reading Statistical Mechanics by R. Pathariya, I came along counting the number of microstates of N-Particle system which are non-interacting and are subject to constraint: $$ \sum^{3N}_{1} \...
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1answer
70 views

The well-defined temperature and 0D Ising model (Ref. Shankar)

I’m reading Shankar’s book Quantum field theory and condensed matter. On page 17, these two bold sentences seem to contradict each other: The system in contact with the heat bath and described by Z ...
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Volume of Plasma as Compared To Gasses

When a material turns from gas to plasma state, does the volume increase, or does it remain the same? And if it does increase, does it depend on temperature and pressure?
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How is energy stored in the surface of a bubble?

In the book of Prigogine, Modern Thermodynamics, it is given on page 147 that Another example is the natural evolution of the shape of a bubble enclosed in a box of fixed $V$ and $T$. In the ...
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Why can't a gas of photons reach a Bose-Einstein condensate?

I have read in many places that as the gas of photons has a chemical potential $\mu=0$ it can't reach a Bose-Einstein condensate (BEC), but I don't understand why. I am reading Greiner's "...
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Can I derive the 1 Dimensional Maxwell-Boltzmann Energy Distribution using an energy sphere?

I was able to deduce the 1 dimensional (1D) MB Speed Distribution of, say $u_z$, by calculating the volume of two rings on a velocity sphere surface at a fixed height $v_z$ then multiply that with the ...
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Free expansion of interacting electron gas is an irreversible process

Consider a number of electron in vacuum, confined in a bounded region. These electrons have interaction described by QED. At $t=0$, we allowed the system to evolve. Without doing any measurement on ...
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Is there a link between the logistic differential equation and Fermi-Dirac statistics?

I was working out some statistical problems and I could not fail to notice that Fermi-Dirac distribution, $$f_{\rm Fermi-Dirac}(E)=\frac{N_{\rm sites}}{1+e^{\beta(E-\mu)}},$$ looks like the kind of ...

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