Statistical Definition and 659 Threads

Hi All, I have recently read about a fallacy that seems to be based on looking at a non-representative subsample of the population. I would like to know if this goes by a name and if it has been formalized. It just seems the problem is that of considering a variable within a subpopulation and...

First, i'd like to apologize for the vague title. Unfortunately my understanding of the question is equally vague. I think the dXd matrix is meant to be a covariance matrix, so the above equation would be some complex constant multiplied by the covariance matrix. The Tr would referring to the...

Why is a thermally isolated process that occurs sufficiently slow is necessarily adiabatic and not just reversible process ? Here I mean that the definition of adiabatic process is no change in the entropy of the subsystem, and a reversible process is define by no change of the total entropy of...

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...

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?

Hi all. Let's assume I have a situation as following. I have a set of x values containing 10 data points. I also got the corresponding measurement values for that each x data points, as y values, and the error on them. Then, I perform calculations, with let's say 5 different models, in where I...

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...

For my own understanding, I am trying to computationally solve a simple spinless fermionic Hamiltonian in Quantum Canonical Ensemble formalism . The Hamiltonian is written in the second quantization as $$H = \sum_{i=1}^L c_{i+1}^\dagger c_i + h.c.$$ In the canonical formalism, the density...

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...

Equations that might be helpful: Attempt: a) (N_max)!/(n!*(N_max-n)!) i.e. N_max C n b) Total Z = sum n=0 to N_max [(N_max C n) e^(buN)] = (1+e^(bu))^N_max Individual Z = 1+e^(bu*1) = (1+e^(bu)) so individual Z^N_max = total Z c) Now, I use Z to represent the total Z, By equation...

I have been browsing this book, and it seems a quite interesting one. The traditional Statistical Mechanics is quite traditionally treated (so only average) but then, the linking of Statistical Mechanics with QFT, and the exact solutions in Conformal Field Theory, are quite nice. But I do not...

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.

<Moderator's note: Moved from a homework forum.> Mass (g) +/- 0.01 grams Drop height (centimeters) +/- 3.00 Shell 53.47 45 No crack 56.78 45 Cracked...

Hi everyone, I have a few questions I'd like to ask regarding what I have read/heard about these two definitions of entropy. I also believe that I have some misconceptions about entropy and as such I'll write out what I know while asking the questions in the hope someone can correct me. Thanks...

Hi all, I have encountered the idea of a heat bath but am slightly perplexed as to what it is. There was a textbook example that looked to find the number density expression for gas molecules as a function of position (image below). It then said that the probability ##P(z)## of finding the...

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...

Homework Statement Consider a system of three aligned spins with S=1/2. There are couplings between first neighbors. Each spin has a magnetic moment ## \vec{\mu} = s \mu \vec{S}##. The system is in a field ## H= H\vec{u_z}## at thermal equilibrium. The hamiltonian is: ##H=J[S(1)S(2)+S(2)S(3)]...

In a lab experiment we had to measure some angles. Every angle measure is the difference between two angular positions and the instrument we used had a resolution of 1', so the uncertainty due to the instrument is $$\sigma_{instr}=\sqrt2'=0.02357... deg$$. We measured the same angle a few times...

Homework Statement We recently collected data on a reverse osmosis system for our unit ops lab. The report LERF (Laboratory Experiment Request Form) requests that we ascertain a value of Aw, the water permeability coefficient, for our membrane. We need to perform a T-Test on the data, however...

Hello, I have a question regarding the derivation for Bose Einstein condensation. I understand that in a boson gas for high temperatures the expectation value of the total number of particles should equal something like: $$ \langle N \rangle \sim T * \eta(z)$$ With ## z = exp(\frac {\mu} {k_b...

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...

Good evening, I have a question to a short introduction to statistical mechanics in a book about molecular dynamics simulation. It introduces the fundamental assumption: Every microscopic state with a fix total energy E is equally probable. I attached the section. I understand it all, except...

Homework Statement [/B] I'm stuck on part (b) and (c) of the following question Homework Equations The Attempt at a Solution The partition function was ##Z_N = 2 cosh(μBβ)## where ##β = \frac {1}{kT}##. From there I used ##U = - \frac {∂}{∂β} ln (Z_n)## to get ##U = -NμB tanh( \frac...

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...

Homework Statement question attached. My question is just about the size of the limit, how do you know whether to expand out the exponential or not (parts 2) and 4)) Homework Equations for small ##x## we can expand out ##e^{x} ## via taylor series. The Attempt at a Solution Solutions...

I apologize if this is not the correct forum for this thread. I have tried to find a discussion regarding this question on the Internet without any success. The Wikipedia discussion https://en.wikipedia.org/wiki/Baryon_asymmetry makes no mention of any statistical explanation, so I understand...

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

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...

Hi I was wondering the following/feeling uneasy about it: Does it make sense to separate the statistical uncertainties of data and Monte Carlo? For example assume infinite statistics in your MC (uncertainty-->0) while your data is finite : so they come with some "uncertainty" (if that makes...

So the pressure for a canonical ensemble is: P = kbT dZ/dV P = pressure P = -∑pi dEi/dV Z = ∑e-βEi pi is the probability of being in microstate i Ei is the energy of state i β = 1/kbT <E> = U = average energy U = -1/Z dZ/dβ = -d(Ln(Z))/dβ How can the pressure (given above) be derived in...

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

I have this word problem, and was wondering how I would go about creating a system of equations. Here is the question: Problem: You are a small forest landowner, and decide you want to sustainably harvest some of timber on your property. There are costs related to the infrastructure needed to...

OK, so, I've forgotten more statistics than my students will ever know, and I'm not too proud to ask for help, because I'm just blanking out on this. I would appreciate it if someone could patiently follow along and let me know what I've got right or wrong please. My understanding of the...

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...

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.

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...

Can anyone refer to some good book on Non-Equilibrium Statistical Mechanics? The book should contain the basics, and can go up to any advanced level. Any level of math is acceptable.

Hello, I was just curious about what academic education is the best to get into statistical physics, and more specifically the statistical physics of optics and lasers. I have considered a few possibilities. Getting a PhD in statistics but take physics electives and most physics courses that...

Sorry, I have never found what does it mean Nuclear statistical equilibrium. It is used in any text but exact explanation nowhere. Please explain a physical meaning of it. Thank you.

Hello, first of sorry for asking what maybe a stupid question. I am teaching myself physics by watching lessons about QM, Classical Mechanics, EMT etc. I was watching Susskind's lectures about statistical mechanics lately and he derived equation of energy E= 3/2 x k x T. 3 in 3/2 came from...

Given a list of values, one calculates the arithmetic mean of those values which are greater than the arithmetic mean of all values. Is there an established name for that quantity?

In our study, we measured the seasonal abundance of selected genes from two sites. We want to make a comparison between the seasonal trends between the genes (i.e. which genes had similar trends and which didn't). What would be the best statistical analysis for this purpose? Thanks!