# Tagged Questions

Statistical mechanics is a branch of mathematical physics that studies, using probability theory, the average behaviour of a mechanical system where the state of the system is uncertain. (Def: http://en.m.wikipedia.org/wiki/Statistical_mechanics)

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### Mathematical calculation

I encountered during my reading to ridge regression that $$(X^TX+\lambda I)^{-1}X^TX = I-\lambda(X^TX+\lambda I)^{-1}$$ What mathematical manipulation has been done here? Thanks in advance
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### Can Fluctuation-Dissipation Theorem Apply to Magnetic Forces in Multi-Spin Systems [closed]

Let's say I have multiple spin systems (atoms in a protein) in a solution of water and the spin systems are all producing a magnetic field $\mathrm{B_{loc}}$ that affects nearby spin systems. Will the ...
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### Pushforward of Liouville measure on configuration space

Say I have $M_1$ a 3-dimensional compact Riemannian manifold, and $M=M_1^{n}$ the product manifold representing n particles on $M_1$. I can identify $TM$ with $T^*M$ via the metric $g$ and then the ...
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### Properties of transfer matrices and their traces

I'm having difficulties understanding some arguments in my statistical mechanics lecture and would like to make them more rigorous by proving some properties. For the Ising model on a lattice we ...
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### Continuously go from a lognormal distribution to a power law

Do you know any phenomena that are described by a continuous mappings between a lognormal and a power law distribution? Of course, one could give a simple linear combination of the two distributions;...
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### Good book on combinatorics for beginners in statistical mechanics

Im studying stat mech and i want to have a better understanding on counting microstates. What book in combinatorics do you guys recommend for beginners like me? Thanks in advance
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### Propagation of Uncertainty from Variance

Given, for a Brownian Motion, the mean square displacement (MSD) of the particles in the 3D are: $MSD_x = t \cdot 10 \mu m/s$ $MSD_y = t \cdot 10 \mu m/s$ $MSD_z = t \cdot 2 \mu m/s$ By using the ...
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### Deriving the q-Gaussian PDF

Ok it may sound a bit too simple but I am quite confused here. While studying generalized entropic forms, in my case that of $S_q$ or in another words the Tsallis Entropy, I reach a point where I have ...
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### Stuck in using Stirling's approximation to show and justify an approximation of the number of permutations with and without ordering

This is a problem from my applied mathematics class where we are currently working on using Stirling's approximation which is: $n! \sim (\frac{n}{e})^n \sqrt{2 \pi n}$ and the context of this ...
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### Expansions of Bose Functions

To study the thermodynamic behavior of the limit $z\rightarrow$ 1 it is useful to get the expansions of $g_{0}\left( z\right),g_{1}\left( z\right),g_{2}\left( z\right)$ $\alpha =-\ln z$ which is ...
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### Statics. Determine the smallest angle $θ$ The coefficient of friction at A and B is $0.5$

Determine the smallest angle θ at which the uniform triangular plate of weight W can remain at rest. The coefficient of static friction at $A$ and $B$ is $0.5.$ Based on my Free body diagram which ...
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### How does the evaluation of $\dot q_i$ at $q_1+\mathrm dq_1$ yield $\dot q_i +\dfrac{\partial \dot q_i}{\partial q_1}\mathrm dq_1 \;?$

I've been following Reif's Fundamentals of Statistical and Thermal Physics; there I came before the derivation of Liouville's theorem: There I couldn't understood few things. I could conceive the ...
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### Average number of steps to return to the origin of a random walk on a 2-d lattice.

Suppose I have a random walker on a 2-d square lattice with periodic boundary conditions with equal probability of going in any of the four directions. The walk ends when the walker reaches the point ...
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### What does $-p \ln p$ mean if p is probability?

In statistical mechanics entropy is defined with the following relation: $$S=-k_B\sum_{i=1}^N p_i\ln p_i,$$ where $p_i$ is probability of occupying $i$th state, and $N$ is number of accessible ...
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### How to expand a factorial expression with $N$ and $m$

In a statistical physics book, I don't understand how they moved from this expression: $\Big(\frac{N}{2}-m\Big)! \Big(\frac{N}{2}+m\Big)! = \Big(\frac{N}{2}!\Big)^2$ with $N=2k, k,m\in \mathbb{Z^+}$,...
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### Special case of the inverse Ising problem with equal correlations

Let $s_1,\dots,s_N\in \{-1,1\}$ be $N$ binary spins. The problem of finding a symmetric interaction matrix $J=(J_{i,j})_{i,j=1}^N$ with zero diagonal and an external magnetic field $h=(h_i)_{i=1}^N$ ...
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### definition of conditional probability $(p_i|\pi_k)$ and Tsallis entropy

Let $\Omega$ be a set of $W$ possible outcomes of an experiment with probability assignments $p_i$ and thus $\sum_{i=1}^{W}p_i=1$. Now, let's divide $\Omega$ into $K$ non-intersecting subsets each ...
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### Logic behind Metropolis algorithm

I am using Metropolis algorithm to make a program for Ising model in Statistical Physics. In Ising model, we take a collection of spins with initial energy, say $E_i$, then we randomly flip one of the ...
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### Probability of Observing $N$ particles in a given volume?

I'm having an issue with a probability problem concerning solutions. Assume there is an "observational region" in a dilute solution with a volume $V$, and as solutes move across its boundary, the ...
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### Metropolis Markov Chain and Mixing Time

I have a statistical mechanical system, which I would like to sample with the Gibbs distribution using a Metropolis-Markov chain. I think the following are standard questions, but I am not sure what ...