Statistical mechanics Definition and 356 Threads

Molecular Transport equations for Liquids are harder to compute than that for gases, because intermolecular interactions are far more important in liquids. A System of equations for particle Distribution function and the correlation functions (BBGKY-Hierarchy) is used in General. For gases, it...

In his well known paper “Information Theory and Statistical Mechanics” Jaynes attempted to formulated statistical mechanics as "nothing more" than the inference theory of many body mechanical systems. I am looking for critiques of this approach. Also of use would be summaries or reviews of the...

With the Liouville's theorem $$\frac{{d\rho }}{{dt}} = \frac{{\partial \rho }}{{\partial t}} + \sum\limits_{a = 1}^{3N} {(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{d{p_a}}}{{dt}} + \frac{{\partial \rho }}{{\partial {q_a}}}\frac{{d{q_a}}}{{dt}})} = 0$$ when we calculate the time evolution...

My professor will be using Huang's Statistical Mechanics next semester and I have been reading a lot of polarizing reviews. Does anyone recommend a book to read parallel to Huang's to better understand the material and that discusses the same topics in similar fashion?

I ran into this kind of expression for a sum that appears in the theory of 1-dimensional Ising spin chains ##\displaystyle\sum\limits_{m=0}^{N-1}\frac{2(N-1)!}{(N-m-1)!m!}e^{-J(2m-N+1)/kT} = \frac{2e^{2J/kT-J(1-N)/kT}\left(e^{-2J/kT}(1+e^{2J/kT})\right)^N}{1+e^{2J/kT}}## where the ##k## is the...

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. Besides the Poisson-Boltzmann equation, I would like to know what are the modern...

## \Omega(E_1)## is the number of microstates accessible to a system when it has an energy ##E_1## and ##\Omega(E_2)## is the number of microstates accessible to the system when it has an energy ##E_2##. I understand that each microstate has equal probability of being occupied, but could...

My first most obvious attempt was to use the relation ##<\epsilon> = \frac{3}{5}\epsilon_F## and the formula for kinetic energy, but this doesn't give the right answer and I'm frankly not sure why that's the case. My other idea was to use the Fermi statistic ##f(\epsilon)## which in this case...

b) Consider P_j(n) as a macrostate of the system, Bosons: P_1(1) = P_2(1) = 1/2*1/2=1/4 ,P_1(2)=P_2(2)=1/2*1/2=1/4 Fermions: P_1(1)=P_2(1)=1 (Pauli exclusion principle), P_1(2)=P_2(2)=0 Different species: P_1(1)=P_2(1) = 2*1/2*1/2=1/2 (because there are two microstates with corresponding to...

I don't know how to solve part c and d. Attempt: c) B_eff=B+e<M> Substitute T_c into the equation in part b, => (B_eff-B)/e = Nμ_B tanh(B_eff/(N*e*μ_B)) Then? Thank you.

I realize the question is quite broad but what research groups working on statistical physics, stochastic processes, and complex systems are generally considered the best? Would like to know about Europe and America alike.

Homework Statement Consider a polymer formed by connecting N disc-shaped molecules into a onedimensional chain. Each molecule can align either its long axis (of length ##l_1## and energy ##E_1##) or short axis (of length ##l_2## and energy ##E_2##). Suppose that the chain is subject to tension...

I have been reading up on the kinetic theory of gases, and I'm unsure whether I have correctly understood why particle velocities become correlated after colliding. Is it because during the collision they exchange momentum and thus their velocities (and hence trajectories) are altered in a...

One version of Liouville’s Theorem for non-dissipative classical systems, governed by a conserved Hamiltonian, is that the volume in phase space (position-momentum space) of an ensemble of such systems (the volume is the Lebesgue measure of the set of points where the ensemble’s density is...

my current skills in math are differential eq and linear algebra... and I am about to start reading Feynman lectures of physics and planning to read all John Baez's recommended books.. after reading Feynman's, what would be the next best thing to do? learn more math? or jump already to core...

Derivation of the Onsager symmetry in many textbooks and papers is as follows: First, assume that the correlation function of two state variables,##a_i## and ##a_j## satifsies for sufficiently small time interval ##t## that $$\langle a_i(t) a_j(0) \rangle = \langle a_i(-t) a_j(0) \rangle =...

Homework Statement ##b_2## is the second virial coefficient Homework Equations Virial expansion: $$P = nkT(1 + b_2 (T)n + b_3 (T)n^2...)$$ $$b_2 = -\frac{1}{2} \int dr f(r) $$ r is the distance vector. $$f(r) = e^{-\beta \phi(r)} - 1$$ The Attempt at a Solution $$b_2(T) = 2 \pi r^2...

Homework Statement A dilute gas of N non-interacting atoms of mass m is contained in a volume V and in equilibrium with the surroundings at a temperature T. Each atom has two (active) intrinsic states of energies ε = 0 and ∆, respectively. Find the total partition function of the gas.Homework...

Hello! Thanks for your time reading my questions. When I was studying quantum statistical mechanics, I get so confused about the relations between Pauli's exclusion principle and the Fermi-Dirac distributions. 1. The Pauli's exclusion principle says that: Two fermions can't occupy the same...

Homework Statement Using log taylor expansion to express cumulants in terms of moments I have worked through the expansion- ##log(1+\epsilon)= ...## see thumbnail- and that's ok; my question is why does the expansion hold, i.e. all i can see is it must be that ##k## is small- how is this...

This is A very general question. I will be taking physics 112 at Cal (in the future) which is basically stat mech. Almost all professors use Kittel and Kroemer but I’ve heard it’s god awful (I can attest to this having read a little myself). Does anyone know of a secondary textbook that is of...

Greetings! It is easy to understand that for a free electron, we can easily define the energy state density, and by doing the integration of the State density* Fermi-Dirac distribution we will be able to figure out the chemical potential at zero kelvin, which is the Fermi-Energy. Hence, we can...

Hello, this is my first question on PhysicsForum. I am primarily interested in statistics/machine learning. I have recently discovered that many of the ideas used in machine learning came from statistical physics/ statistical mechanics. I am just wondering if it's a bad idea to attempt to learn...

At my school, you have to take Quantum mechanics at the same time as Statistical mechanics (co-requisites) in either junior or Senior year as a physics major; why is this? What is the relationship between the 2?

I was told that defining temperature as the "average kinetic energy of the particles in a system" is not accurate enough.

Homework Statement A polymer chain consist of a large number N>>1 segments of length d each. The temperature of the system is T. The segments can freely rotate relative to each other. A force f is applied at the ends of the chain. Find the mean distance ##\textbf{r}## between the ends...

I’ve been reading up about Boltzmann transport equations, and the concept of molecular chaos has come up, in which one assumes the velocities of particles are assumed to be uncorrelated. I’m a bit confused about the concept though. In what sense do the velocities become correlated in the first...

Homework Statement Determine if the classical approximation (Maxwell-Boltzman statistics) could be employed for the following case: a) Electron gas in a metal at 2.7K (cubic metal lattice of spacing 2Å) Homework Equations Maxwell-Boltzman statistics are acceptable to use if the de broglie...

Homework Statement Calculate the change in molar entropy of steam heated from 100 to 120 °C at constant volume in units J/K/mol (assume ideal gas behaviour). Homework Equations dS = n Cv ln(T1/T0) T: absolute temperature The Attempt at a Solution 100 C = 373.15 K 120 C = 393.15 K dS = nCvln...

Homework Statement ∫ e1000((sinx)/x) dx [0 to 1000 : bound of integration]. Solve this integral of a sharply peaked function without a calculator. Homework Equations I'm doing this in relation to statistical thermodynamics - I think I need to use Sterling's Approximation or a gamma function...

In general when one talks about an idealized gas, should/could one assume it is non-relativistic? (s.t E=p^2/2m will hold) many thanks

Homework Statement I'm working on a process similar to geometric brownian motion (a process with multiplicative noise), and I need to calculate the following expectation/mean; \langle y \rangle=\langle e^{\int_{0}^{x}\xi(t)dt}\rangle Where \xi(t) is delta-correlated so that...

Hi all. Where can I find some good introductory sources teaching the use of statistical mechanics to study things like tornado formation or climate in general? I took P. Chem., a while ago now but I'm reviewing the material independently. We used one of Moore's texts, 80's - ish, and in it he...

Question Form the canoncial partition using the following conditions: 2 N-particles long strands can join each other at the i-th particle to form a double helix chain. Otherwise, the i-th particle of each strand can also be left unattached, leaving the chain "open" An "open" link gives the...

Long time no see, PhysicsForums. Nevertheless, I have gotten myself into a statistical mechanics class where the prof is pretty brutal and while I can usually manage, this problem finally has me stumped. I'd like to be nudged in the right direction, not outright given the answer if possible. I...

Hello - I am wondering abut the meaning of chemical activity. Most definitions are something along the lines of "effective concentration," which is fine until you have a real concentration in a lab, and you don't know if you need to calculate the concentration or "effective" concentration of the...

I'm working on a project at university to calculate the magnetocaloric effect of dysprosium. This will be done using a new technique designed at the university of which its not necessary to go into detail about. In short, the Dy is placed in a solenoid, through which a current runs, the current...

1. Homework Statement Question attached. I am looking at the second line limit ##\beta (h/2\pi) \omega << 1 ## 2. Homework Equations above 3. The Attempt at a Solution Q1)In general in an expansion we neglect terms when we expand about some the variable taking small values of the...

Greg Bernhardt submitted a new PF Insights post Statistical Mechanics Part II: The Ideal Gas Continue reading the Original PF Insights Post.

I am trying to understand the derivation for the DOS, I get stuck when they introduce k-space. Why is it necessary to introduce k-space? Why is the DOS related to k-space? Perhaps if someone could come up to a slightly different derivation (any dimensions will do) that would help. My doubt ELI5...

Max Planck formulated the quantum hypothesis, that electromagnetic radiation was emitted from heated bodies only in quanta of energy E = hf, where f was the frequency of the radiation and h was a constant now called “Planck's Constant”, in order to solve the Ultraviolet Catastrophe...

Homework Statement Consider the system of two large, identical Einstein solids, each with oscillators, in thermal contact with each other. Suppose the total energy of the system is 2 units of the energy quanta, i.e., =2ℏ, (i) how many MACRO-states (e.g., one macro-state corresponds to one...

My question is regarding a few descriptions of Entropy. I'm actually unsure if my understanding of each version of entropy is correct, so I'm looking for a two birds in one stone answer of fixing my misunderstanding of each and then hopefully linking them together. 1) A measure of the tendency...

Hello, I'll try to get right to the point. Why and how does logarithmic dependence appear in statistical mechanics? I understand that somehow it is linked with probabilities, but I can not quite understand.

Homework Statement Is the statement ”Given a two-fermion system, and two orbitals φ labeled by quantum numbers a, b, the two-body wavefunction (1,2 represent the particle variables) $$\psi(1,2) = \phi_a(1) \phi_a(2) - \phi_b(1) \phi_b(2) + \phi_a(1) \phi_b(2) - \phi_b(1) \phi_a(2) $$...

Hi! How do I check if the equation of motion of the particle, with a given potential, is invariant under time reversal? For a 2D pointlike particle with potential that is e.g $$V(x) = ae^(-x^2) + b (x^2 + y^2) +cy', where a,b,c >0$$ Can it be done by arguing rather then computing? Thanks!

Greg Bernhardt submitted a new PF Insights post Statistical Mechanics Part I: Equilibrium Systems Continue reading the Original PF Insights Post.

In statistical mehcanics(pathria, 3rd edition), I have some questions for ideal fermi and bose gases. The textbook handles the approximation for z(=e^βµ) and nλ^3 (n=N/V, λ : thermal de Broglie wavelength). It considers the cases that z<<1, z~1, nλ^3~1,<<1,→0 and so on. In here, I am confused...

Hi guys, I have been reading some of the literature recently concerning the Kardar-Parisi-Zhang equation and the words "universality" and "KPZ universality class" keep appearing. I already did a cursory wikipedia search on the subject, but it did not make much sense to me. Can you please...

While doing some calculations on v_rms using the Maxwell-Boltzmann distribution, I noticed that v_rms and v_avg are pretty similar (https://casper.berkeley.edu/astrobaki/index.php/File:MaxwellSpeedDist.png). In fact, really it's just the choice of using the 1-norm (|v|_avg) vs. 2-norm sqrt(v^2...