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.
Looking to evaluate an integral of the form $$\int_0^{\infty} \frac{p^2 dp}{\mathrm{exp}(a\sqrt{p^2+b^2}) \pm 1} $$Changing to ##x(p) = a\sqrt{p^2 + b^2}## gives $$\frac{1}{a^3} \int_0^{\infty} \frac{\sqrt{x^2-(b/a)^2}}{e^x \pm 1} dx$$Wolfram alpha doesn't tell me anything useful, sadly.
So.. question:
- How do I know that only the pin is at work at E and not those 2 beams? my guess: It is because those 2 beams are connected to the pin whilst the pin is the one that exerts a force on that walking beam DEF?
I found this little book titled “Statistical Mechanics; A set of lectures” by Feynman in the library. I’m not taking Stat Mech until Easter so I’d just be reading for interest at this stage, although the content looks fairly involved. Is it suitable for a first introduction?
So just by by using the definition of the partition function...
$$ Z = \sum_i e^{ \frac {-E_i} {k_BT} } = e^{ \frac {-0} {k_BT} } + e^{ \frac {-\epsilon} {k_BT} } = 1 + e^{ \frac {-\epsilon} {k_BT} } $$
And then, a result we obtained in class by using the Boltzmann H factor to solve for ##S##...
My attempt : $$P(n) = \frac{1}{\mathcal{Z}} Exp[(n\mu -E)/\tau]$$, use $$\lambda = e^{\mu/\tau}$$, then the distribution can be written as $$P(n) = \frac{1}{\mathcal{Z}} \lambda^nExp[-E/\tau]$$
Note that the average number of particle can be written as $$<N>= \lambda \partial \lambda ( log...
It even gives a hint, it says "consider two horizontal surfaces z1 and z2 and think about what thermodynamic equilibrium means for particles traveling from one surface to the other". This really trips me up because I am not sure what to do with this. Obviously in equilibrium the number of...
I was reading mehran kardar (books and lectures) it says the concept of irreversibility comes from an assumption (in which we increase the length scale by interaction disctance between two particles).
So My question is the concept of irreversibility is still valid in the case of 1 particle...
I am an incoming graduate student in Theoretical Physics at Universiteit Utrecht, and I struggle to make a choice for one of my mathematical electives. I hope someone can help me out. My main interests lie in the fields of Statistical Physics, phase transitions and collective and critical...
Early in Bellan he asks us to consider a finite-temperature plasma and assume that the ion and electron densities are initially equal and spatially uniform. He approaches the problem of calculating the Debye length by considering each species of particle, σ
, as being a fluid so that the...
Problem Statement: An explosive liquid at temperature 300 K contains a spherical bubble of radius 5 mm, full of its vapour. When a mechanical shock to the liquid causes adiabatic compression of the bubble, what radius of the bubble is required for combustion of the vapour, given that the vapour...
Not sure this is the right area to post this.
Let's say I have particles on a lattice, and they all have some property (ie, color) that is correlated at some known correlation length. I want to simulate this! In 1D I could do something like have color be a random walk in the given dimension...
Hello! I have a question about the first problem in Stat Mech (Physical Adsorption) from http://web.mit.edu/physics/current/graduate/exams/gen2_F01.pdf. The solution to it can be found http://web.mit.edu/physics/current/graduate/exams/gen2sol_F01.pdf. I understand the logic they use for the...
Homework Statement
Show that the FD distribution can be viewed as giving the probability that a given state ( of the prescribed
energy) is occupied.
Homework EquationsThe Attempt at a Solution
Solution to this problem:
I understand the solution,but I took a different approach...
Homework Statement
Homework EquationsThe Attempt at a Solution
So cts approx holds because ##\frac{E}{\bar{h}\omega}>>1##
So
##\sum\limits^{\infty}_{n=0}\delta(E-(n+1/2)\bar{h} \omega) \approx \int\limits^{\infty}_{0} dx \delta(E-(x+1/2)\bar{h}\omega) ##
Now if I do a substitution...
I just started learning some Stat. Mech by Leonard Susskind's lectures and am in a part briefly overviewing basic probability in general; One of the things brought up was F(i), some quantity associated with the ith state of a system, and the important average of F(i) averaged over the...
1. The problem statement, all
variables and given/known data
Show that the probability density as a function of seperation, r, of two atoms interacting via a potential U(r) (e.g. a function of separation only, such as a Coulombic interaction), is given by
$$\rho(r) = Cr^2e^{-\beta U(r)}$$...
Hey guys, I'd like to ignore Hamiltonian mechanics and use a specific example of the many particle Schrodinger equation in Griffith's chapter 5.4 to try to get to statistical mechanics, and see what happens, I need help making sense of it:
Given 3 non-interacting particles in a 1-d potential...
Homework Statement
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Why does it take longer to cool water in a refrigerator than it takes to cool air, assuming that the inside of the refrigerator is initially at the same temperature as the room it is in?
Homework EquationsThe Attempt at a Solution
Quite frankly I don't even know if my...
Homework Statement Consider a system of 2 non interacting distinguishable particles in thermal equilibrium at temperature T, and which as two possible energy states available: E1 and E2>E1.
How would you go about finding the average number of particles in energy level E1, and in hight T limit...
Homework Statement
Suppose that particles of two different species, A and B, can be chosen with
probability p_A and p_B, respectively.
What would be the probability p(N_A;N) that N_A out of N particles are of type A?
The Attempt at a Solution
I figured this would correspond to a binomial...
Hello,
I've forgotten much of what I learned in my undergraduate thermal physics class. This is unfortunate, as my statistical mechanics class just builds on those concepts, especially the differential relationships between various state variables. I was wondering if someone could link me to...
Homework Statement
Prove for the canonical ensemble
##\overline{(E-U)(p-\overline{p})}=kT[(\frac{\partial U} {\partial V})_{N,T} + p]##
Left hand side is covariance
E is energy
U is internal energy, average of E
p is microstate pressure
$$\overline{p}$$ is average pressure
V is volume
N is...
Greetings,
This question makes reference to the stat mech book, “Fundamentals of Statistical and Thermal Physics”, by Reif, so people who have that book will probably understand where I am coming from most easily. However, the main points/questions of this post are independent of the book, so...
As the title says would it be an overload to take all those classes together? I am wondering because these classes are only offered in the fall semester and I plan on taking the physics subject GRE next fall. What should I expect? Any input would be helpful thank you!
Hi,
I will start working on stat mech problems with use of Monte Carlo methods for a summer internship. I don't have more details about what I'll be doing for now but I'm already looking for resources for becoming familiar with the tool. I already used Monte Carlo algorithms to compute high...
I'm an engineering physics major, I believe I have a way to graduate (if I jettison some electives and take the minimum necessary to graduate) at the end of Spring semester 2014 (or maybe extend things into summer), and I take stat mech in Fall 2013. Grad school apps are due Dec/Jan/Feb, I am...
Homework Statement
Consider a system of N distinguishable, non-interacting harmonic oscillators. The Hamiltonian is given (shown below).
Assuming that the oscillators obey Schrodinger's equation, determine the canonical partition function for the system. Then assume the oscillators obey...
deciding between 2 grad courses next semester, solid state and stat mech. I've already taken the undergrad versions of both. Got an extremely low grade in solid state the first time, A in stat mech. Currently grad student in condensed matter physics.
The solid state class will focus on...
Homework Statement
I have come across a rather interesting conundrum. Given the configurational potential energy partition function for a non-ideal gas:
Z = Z_{internal}\frac{1}{N!}\left(\frac{2\pi m}{h^{2}\beta}\right)^{\frac{3N}{2}}(V^{N} - B_{2}(T)N^{2}V^{N-1})
where B_{2}(T) is the...
Homework Statement
Homework Equations
The Attempt at a Solution
I have a vague idea of how to do this problem, but I'm not sure. Here's the plan I have for solving it:
1. Find the dispersion relationship \epsilon(k)
2. Find the 2D density of states for wave vectors k: g(k)dk
3. Using the...
Homework Statement
It's easier to post a picture of the problem:
Homework Equations
In picture, and boson occupation number:
\left\langle n_k \right\rangle = \frac{1}{e^{\beta E(k)} - 1}
Where E is the energy of the state with k and \beta = 1/k_B T
The Attempt at a Solution
Goal: Find...
Problem solved.Homework Statement
The boiling point of a certain liquid is 95OC at the top of a mountain
and 105OC at the bottom. Its latent heat is 1000 cal/mole. Calculate the
height of mountain.
It's Q4 of chapter 15 in Statistical Mechanics by S.K. MaHomework Equations
it should be the...
Homework Statement
You are told that you have 1 mole of an ideal gas with heat capacity at constant volume being 1.5R and you send it over an arbitrary path where dq/dt|pathway= 2R. In the end, the volume of the gas doubles, so figure out by what factor the temperature must change. Assume...
Hi, I was wondering if someone could point me to a textbook or easy to read paper (or website) that briefly describes/proves the differences here. What I mean is if I do classical (continuous energies) statistical mechanics where my initial state is a volume (greater than or equal to h-bar)...
Homework Statement
I already have the solution to the problem. Just need some help deciphering the logic behind it.
There are V balls, which are identical except for their color. N of them are blue and V-N are red. We place the balls inside a box so that V/2 balls are on each side...
Hello,
Given equilibrium, why does one only consider a mixed state where the pure states are eigenfunctions of the hamiltonian, i.e. states with an energy eigenvalue?
And for the second question, I quote David Tong's ``Lectures on Statistical Physics'' ( freely and legally accessible on...
Homework Statement
I'm trying to use the transfer matrix method in statistical mechanics but I'm struggling with the algebra so I'd like to know if there is a simpler way to find the eigenvalues and eigenvectors of a matrix.
For example, studying the lattice gas model produces the transfer...
Homework Statement
Consider a system of lightlike particles in two dimensions with energy spectrum /omega = ak^3. What is the specific heat capacity at low temperatures?Homework Equations
The Attempt at a Solution
Is this as simple as Z = /integral from -inf to inf of exp[-/beta /omega] d...
So, when people speak of "statistical physics", they're usually referring to "statistical mechanics". But statistical mechanics is really just a small subset of all of statistical physics (unfortunately, that makes googling for statistical physics difficult, because you'll mostly get stat mech...
Homework Statement
A system has a two component order parameter: [phi1, phi2]. The mean-field Landau free energy is E(\phi_1,\phi_2) = E0+at(\phi_1^2+\phi_2^2)+b(\phi_1^2+\phi_2^2)^2+c(\phi_1^4+\phi_2^4)
where t, a, and b are positive constants.
Represent the order paramter as a vector on...
So lately I've been thinking about not going to graduate school and getting a job after undergrad instead. However, at this point I'm still really unsure what I want to do so I'm fairly certain that I'm still going to apply to graduate school. Next semester I was planning on taking stat mech...
Homework Statement
lim_{dt\rightarrow 0} [(1+ \alpha dt(e^{-ik}-1))^{1/dt}]^T = e^{\alpha (e^{-ik-1)T}
It's a well known property in statistical physics, I'm not sure how to prove it
Homework Equations
The Attempt at a Solution
I know dt ->0
and 1/dt -> infinity
Which one converges...
An ideal gas of N molecules in a volume V. Write down the mean and standard deviation of the number of molecules n to appear in a volume V/3. Sketch the probability distribution of n and of n/N. Indicate how the curves change as N increases.
ok. \bar{n}=\sum_i n_i P_i
do i use P_i form the...
Homework Statement
I don't want to do it word for word, but its like this:
Statistical Mechanics is the topic.
Given a volume V and N particles within it.
Assume no correlation in the location of the particles.
1. Find the probability that a region of volume "v" contains "n" exactly...
i just finished my analytic mechanics and am halfway done with upperdiv E&M. Next fall, i plan on taking the 2nd half of E&M and the 1st quarter of Quantum and Abstract Algebra, for sure. Along with that, I'm thinking of taking either Statistical Mechanics or Solid State Physics.
at my...
Homework Statement
My stat mech book does a problem where it calculate this quantity F for a system of particles restricted to move in one dimensions using the equation F = \frac{\partial A}{\partial L} where A is the helmholtz free energy. What I am confused about is that I thought F...
Hi,
I've acquired a few stat mech texts.
1)Landau & Lif****z: A Course in Theoretical Physics: Statistical Mechanics part 1 and 2
2)Huang
3)Chandler
Which of these should I start with to self-study statistical mechanics? I'm eager to read one of the famous Landau texts, but I'm afraid...