What is Statistical mechanics: Definition and 393 Discussions
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles.
Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties such as temperature, pressure, heat capacity, in terms of microscopic parameters that fluctuate about average values, characterized by probability distributions. This established the field of statistical thermodynamics and statistical physics.
The founding of the field of statistical mechanics is generally credited to Austrian physicist Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates, to Scottish physicist James Clerk Maxwell, who developed models of probability distribution of such states, and to American Josiah Willard Gibbs, who coined the name of the field in 1884.
While classical thermodynamics is primarily concerned with thermodynamic equilibrium, statistical mechanics has been applied in non-equilibrium statistical mechanics to the issues of microscopically modeling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions or flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.
Is there some kind of resolution to the Hydrogen atom problem in statistical physics, that is the fact that canonical partition function diverges for E_n = - E_0/n^2 with degeneracy n^2 since Z = \sum n^2 exp(-\beta E_0/n^2) > \sum n^2 exp(-\beta E_0) , which makes the H atom problem seem...
Just FYI, this is not a homework question. I am working on a practice problem for my final exam and I decided to post it here since I think it would get more attention than in any other section.
Homework Statement
Given an unbranched polymer with N links (that can be oriented in any...
I'm kind of stuck on this problem, if someone could help me out that would be appreciated.
Consider 2 blocks treated as collections of Einstein oscillators. The first block has N1 oscillators of frequency omega. The second block has N2 oscillators of frequency 2omega. Initially the first...
Statistical Mechanics preparedness...?
Hello, I am a third year physics major having trouble deciding wether to take Statistical Mechanics this Spring semester. The problem is that there are 2 pre-requisites, Quantum Mechanics 1 and Thermodynamics, and I am only taking Thermodynamics this...
Hi all,
It's my first time to ask a question here
I am now taking a Thermodynamics course and I have the authority to choose what topics to study in this course.
My intention is to be able to study statistical mechanics afterward. So I need thermodynamics that will be useful when studying...
hi everyone!
I am so confused about interaction energy!
as you know, in many statistical mechanics books we see there are equations which are taken in a non-interacting system.
for example for a two particle system, you can see the total energy is:
E = E1 +E2 +E12
while E12 is the...
Homework Statement
Spectroscopic data (rotational-vibrational lines) show that the hydrogen fluoride molecule has a vibrational frequency of 7.8x10^14 rad/sec and a moment of inertia of I=1.35x10^-47 kg.m^2. Find the relevant characteristic temperatures of the HF molecule.
Homework...
So I'm not asking if thermodynamics has, but specifically statistical mechanics, because it seems so unlikely that by probabilistic and mechanical reasoning, the phenomenon of self-organization would arise -- and if it has happened, I'd love to find out. I'm not looking for specific papers, as I...
I am trying to work out the heat capacity of a body-centered cubic iron lattice using stat.mech., but am having some trouble.
Firstly, I assumed that the iron atoms behaved as harmonic occilators, not taking electronic or nuclear spin into account. Is this a good or bad approximation?
Then...
Hello, so I was watching Susskind's lectures on Statistical Mechanics:
He explained Liouville's Theorem qualitatively in the following two ways:
No Merging: Two trajectories in phase space will never merge; this seems obvious using the time-symmetry and determinism in classical mechanics...
Hello,
I've finished the topics about statistical mechanics in Feynman Lectures,and I'd like to study the topic in a deeper aspect. I've had a look at Berkeley Course vol 5(by Reif),I liked it,but I find it rather too introductory. I also had a look at Huang,but I think it is too advanced for...
We're covering probability of the distance for free electrons with parallel spin (long-range oscillations should go to zero) and using that to get a correlation energy. My teacher wants us to elaborate the following 1D case.
\int e^{ik(x-X)}dk=\frac{e^{ik(x-X)}}{i(x-X)}\Rightarrow...
The Constraint Based Statistics --- Beyond the Entropy Based Statistical Mechanics
The Constraint Based Statistics --- Beyond Tsallis Entropy and Boltzmann Entropy Based Statistical Mechanics
This post is a summary about a brand new work in the field of Nonextensive Statistical Mechanics...
I have some questions that my teacher was unable (and unwilling) to answer in class, so I thought I'd ask them here.
The chemical potential
\mu=\left(\frac{\partial U}{\partial N}\right)_{S,V}
is given by the derivative of the energy with respect to particle number at constant volume and...
Homework Statement
In a system of N weakly interacting particles each particle can be in one of M energy states:
E_1 < E_2 < ... < E_M
At T=300K there are 3 times as many particles in E_2 as in E_1.
Calculate E_2 - E_1
Homework Equations
This is not my homework, just a tutorial question, I'm...
In derivation of probability of system at energy E with canonical ensemble, one assumes that the probability of system in a microstate Ei is proportional to the multiplicity of reservoir. Is this probability the conditional probability by knowing that system is at energy Ei with knowing it is at...
I took stat mech as an undergrad but the textbook we used (statistical and thermal physics by sturge) was over my head. Can someone provide a good and readable (as readable as stat mech can get) textbook for stat mech? I am switching to a different research group in grad school that deals with...
When I learned statistical mechanics, it followed the lines of the maximum entropy principle from information theory as laid out by Jaynes which can also be seen as a Bayesian statistical theory.
I wonder whether there exist some orthodox frequentistic interpretation of statistical mechanics...
I was thinking of taking the statistical mechanics course for PhD candidates that's being offered next term at my school. My background is pretty typical, I've had Calc 1-3, linear algebra, differential equations, classical mechanics, e&m, thermodynamics with statistical mechanics, all at the...
Homework Statement
a) Derive an asymptotic expression for the number of ways in which a given energy E can be distributed among a set of N, one-dimensional harmonic oscillators, the energy eigenvalues of the oscillators being (N+\frac{1}{2})\hbar\omega, n=0, 1, 2, ....
b)Find the corresponding...
One postulate of sta mech is all accessible microstates are equally probable in thermal equilibrium, while according to classical mechanics this postulate is not true. But the postulate is still applicable in the real world. I remember that I read one interpretation before:
1. any particle in a...
Homework Statement
Consider a particle confined within a box in the shape of a cube of edges Lx=Ly=Lz. The possible energy levels of this particle are then given by the quantized energy for a particle in a 3D box.
Calculate explicitly the force per unit area (or pressure) on this wall...
Homework Statement
Consider a particle confined within a box in the shape of a cube of edges Lx=Ly=Lz.
(a) Suppose that the partice is in a given state specified by particular values of the principal quantum numbers nx, ny, nz. By considering how the energy of this state must change when...
I am currently enrolled in PHY4523 - Thermodynamics and Statistical Mechanics where we are currently using "Fundamentals of Statistical and Thermal Physics" by F. Reif and I was wondering if there were any good recommendations as far as supplemental texts that I could use in helping me...
Homework Statement
From Landau and Lifgarbagez:
\langle (\Delta f)^{2} \rangle = \overline{f^{2}} - (\overline{f})^{2}
This isn't derived, just stated, and I'd like to understand how it comes about. f is a generic quantity "relating to a macroscopic body or to a part of it."...
I posted this once already without seeing the rule that HW questions must go here--sorry :redface:
So, the problem: I'm really lost on where to get started here. It's a two state system, one with energy 0 and the other with energy ε. I already have ensemble average, <E>, found to be:
ε / (e^βε...
I'm really lost on where to get started here. It's a two state system, one with energy 0 and the other with energy ε. I already have ensemble average, <E>, found to be:
ε / (e^βε + 1) , where β is thermodynamic beta, 1/KbT.
How do I convert this to an expression for the variance of the...
So I didn't start my physics major until my sophomore year which means I'm a year behind. Because of this I won't be able to take quantum mechanics until the first semester of my senior year which is also when statistical mechanics is offered. I'd really like to take statistical mechanics but...
Question 1:
The mean speed for escaping molecules from a hole is 1.88*(k*T)/m)^1/2 (Eq.1)
The mean speed for molecules in the Maxwell-Boltz Distribution is 1.596*((k*T)/m)^1/2 (Eq.2
If you were to calculate molecular flux for an ideal gas, which expression would you use?
In my...
Consider an ideal gas of oxygen atoms in equilibrium with oxygen atoms absorbed on a planar surface. here are N_s sites per unit surface area at which the atoms can be absorbed, and the energy of an absorbed atom is -e compared to one in the free state. The system is under 1 atm and at 300K...
I am quite well versed with the random walk problem and am interested in finding out more about Brownian motion. Does anyone have any suggestions for books that explain Brownian motion in detail? I suspect these will be books on statistical mechanics.
Hi! I'm a graduate student in solid state physics and I have to follow a graduate course on equilibrium statistical physics, and we're using Plischke and Bergersen's book on "Equilibirum Statistical Physics". Presently, we're seeing Mean Field Theories and Ising model, but somehow, I'm not...
Given a macro-state M of a system, let S denote the potion of the phase space that has the macro-state M.
A micro-canonical ensemble is one in which the probability of finding the micro-state in any part of S is equally likely (the density function is constant over S).
In Pathria's...
I'm not sure if this is the proper location for this thread, but its for a math course and I think my issues concern the math portion of the problems. If it should be moved please do so.
Note: I know the post is quite long, so I'll just pull out my few main issues from all the junk.
1)...
I've been refreshing myself on some of the statistical mechanics I learned a couple years ago, using Kittel and Kroemer as a guide. However, I've come across a couple things that bother me:
1. When the Boltzmann distribution is derived, no real physics enters the picture. Essentially, the...
Hi guys, I am reading Reif's book on statistical mechanics and have a question on the H theorem. In section 2.3, Reif gives (on page 54) both the definition of equilibrium as well as a fundamental postulate.
Definition: "An equilibrium situation is characterized by the fact that the...
I have of late been reflecting on something.
Generally as a rough approximation we may divide physics into classical mechanics, quantum mechanics, classical field theory (like E/M, fluid mechanics...), quantum field theory, and then statistical mechanics.
All the classical and quantum...
I am currently trying to get my head round RG in the context of statistical mechanics and am not succeeding! I would be grateful for any help. I have a specific question, but any clarification of RG in general would be useful. Here is my understanding of the main ideas:
1. Define...
I have my main UG exam on Sat & i dnt have answer to these questions . Help me out
Q. Distinguish b/w degenerate energy level & a degenerate gas ?
Ans : Degenrate level : A single energy level can be degenerate with another energy level , no difference b/w their energies . Degenerate...
Homework Statement
N weakly interacting distinguishable particles are in a box of volume V. A particle can lie on one of the M possible locations on the surface of the box and the number of states available to each particle not on the surface (in the gas phase) is aV, for some constant a.
1...
Homework Statement
Using the microcanonical ensemble, find the entropy for the mixing of two ideal gases, but we need to compute all at once instead of separately for each gas and adding the two.
Homework Equations
\Omega(E)=\int_{H<E}d\overline{p}d\overline{q}
S(E,V)= k_{B}...
http://www.ph.ed.ac.uk/~pmonthou/Statistical-Mechanics/documents/SM15.pdf
At the top of page 59 in the above link, I can't see where the first three bullet points that follow
E(T=0)=\int_0^{\epsilon_f} g(\epsilon) d \epsilon
come from?
can anyone help?
In my notes,
http://www.ph.ed.ac.uk/~pmonthou/Statistical-Mechanics/documents/SM3.pdf
on page 1 we are told we're dealing with systems of fixed total E but in the expilicit example on page 3, do we not find 4 different values for the total energy. how is this possible?
I have to take that next year and am nervous about it. I took it as an undergraduate, but most of it went right over my head. Now I have to take the graduate-level course. Can anyone recommend a good book for me to read in preparation for that class?
In adjacent containers of volume V1 and V2, there contains a gas of N molecules. The gas is free to move between the containers through a small hole in their common wall.
What is the probability to find k molecules in the V1?
Probability of one molecule to be in V1 is given by...
I am trying to show that the change in number of accessible microstates, and therefore the change in the multiplicity function g of a simple system is
g=\left( \frac{\tau_F^2}{\tau_1\tau_2}\right)^{\frac{mC_V}{k_B}}
where the system is two identical copper blocks at fundamental...
Imagine a jet of fluid (perhaps air) impinging on a flat plate. It could be said that the jet has a slightly higher mean velocity in the direction normal to the flat surface (we'll arbitrarily call this X).
From a classical thermodynamic point of view it could be said that the gas has a higher...