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数学代写|7CCM314B统计力学代写 Statistical mechanics代写

Part A

Consider a random variable $X$ taking values in the sample space $\Omega_{X}={0,1,2,3, \ldots}$. Denote by $p_{j}=\operatorname{Prob}[X=j]$ with $j \in \Omega_{X} .$ Let $Y$ be a Poisson random variable with parameter $\lambda$, defined over the same sample space $\Omega_{X}$
$$q_{j}=\operatorname{Prob}[Y=j]=\frac{\lambda^{j} \mathrm{e}^{-\lambda}}{j !}$$
(a) [5 points] Show that the distribution $\left{q_{j}\right}$ is correctly normalised over $\Omega_{X}$.
(b) [ 5 points] Show that the average of $Y$ is equal to $\lambda$.
(c) [ 7 points] Compute the KL-divergence $D_{\mathrm{KL}}(p | q)$ between the two distributions as a function of $\lambda$. The result should include the average $\langle X\rangle=\sum_{j=0}^{\infty} j p_{j}$ and the entropy $H(X)$ of $X-$ among other term(s).
(d) [3 points] Find the minimum of $D_{\mathrm{KL}}(p | q)$ by differentiating with respect to $\lambda$. Deduce which value of the Poisson parameter $\lambda$ must be chosen to best approximate any distribution over the non-negative integers with a Poisson distribution.

Consider a random variable $X$ taking values in the sample space $\Omega_{X}={0,1,2,3, \ldots}$. Denote by $p_{j}=\operatorname{Prob}[X=j]$ with $j \in \Omega_{X} .$ Let $Y$ be a Poisson random variable with parameter $\lambda$, defined over the same sample space $\Omega_{X}$
$$q_{j}=\operatorname{Prob}[Y=j]=\frac{\lambda^{j} \mathrm{e}^{-\lambda}}{j !}$$
(a) [5 points] Show that the distribution $\left{q_{j}\right}$ is correctly normalised over $\Omega_{X}$.
(b) [ 5 points] Show that the average of $Y$ is equal to $\lambda$.
(c) [ 7 points] Compute the KL-divergence $D_{\mathrm{KL}}(p | q)$ between the two distributions as a function of $(a) [8 points] State and prove the inequality part of Jensen’s theorem for the case of an alphabet of two outcomes,$\mathcal{A}{X}=\left{a{1}, a_{2}\right}$, with probabilities$\mathbb{P}\left(X=a_{1}\right)=p_{1}$and$\mathbb{P}\left(X=a_{2}\right)=p_{2} .$Do not assume any specific form for$f(x)$in the proof. (b) [ 6 points] Use Jensen’s inequality with the convex function$f(x)=-\ln x$to prove the inequality of arithmetic and geometric means, namely for any$n$non-negative real numbers$x_{1}, \ldots, x_{n}$$$\frac{x_{1}+x_{2}+\ldots+x_{n}}{n} \geq\left(x_{1} x_{2} \cdots x_{n}\right)^{1 / n}$$ (c) [6 points] Use Jensen’s inequality with the convex function$f(x)=\sqrt{x^{2}+1}$to prove that for all$n \in \mathbb{N}$$$\sqrt{1^{2}+1}+\sqrt{2^{2}+1}+\ldots+\sqrt{n^{2}+1} \geq \frac{n}{2} \sqrt{n^{2}+2 n+5}$$ 问题 3. Consider a random variable$X$taking values in the sample space$\Omega_{X}={0,1,2,3, \ldots}$. Denote by A charitable organisation is organising one raffle per day over a period of$N$days to raise money for the environment. Assume that before the$j$-th draw, a given person can buy$\sigma_{j}$tickets, with$\sigma_{j}$a random variable defined on the sample space$\Omega={0,1,2, \ldots, 8}$. Let$\sigma=\left(\sigma_{1}, \ldots, \sigma_{N}\right)$be the sequence of tickets purchased by this person prior to each of the$N$draws. Assume that$N$is even. Let $$M_{O}(\boldsymbol{\sigma})=\sum_{\text {odd } j} \sigma_{j} \quad M_{E}(\boldsymbol{\sigma})=\sum_{\text {even } j} \sigma_{j}$$ be the total number of tickets purchased during odd-numbered and even-numbered days, respectively. Assume that we can reliably estimate the average values$\left\langle M_{O}(\boldsymbol{\sigma})\right\rangle$and$\left\langle M_{E}(\boldsymbol{\sigma})\right\rangle$over many raffle seasons. In the following, you may express the results of calculations using the shorthands$\varphi(\lambda)=\sum_{n=0}^{8} \mathrm{e}^{\lambda n}$and$\theta(\lambda)=\sum_{n=0}^{8} n \mathrm{e}^{\lambda n}$, without evaluating the sums explicitly. (a)$[6$points$]$By finding the minimum of a suitably defined Lagrangian$\mathcal{L}$, determine the probability distribution$P(\sigma)$(depending on two Lagrange multipliers$\left.\lambda_{1}, \lambda_{2}\right)$that maximises the entropy$S[P]=-\sum_{\sigma} P(\sigma) \ln P(\sigma)$subject to the constraints on$\left\langle M_{O}(\boldsymbol{\sigma})\right\rangle$and$\left\langle M_{E}(\boldsymbol{\sigma})\right\rangle$. Determine the partition function explicitly. (b) [2 points] Express the free energy$F$in terms of the Lagrange multipliers$\lambda_{1}, \lambda_{2}$(c) [2 points] Using the free energy$F$, determine$\left\langle M_{O}(\boldsymbol{\sigma})\right\rangle$and$\left\langle M_{E}(\boldsymbol{\sigma})\right\rangle$as functions of the Lagrange multipliers$\lambda_{1}, \lambda_{2}$(d) [4 points] Using the results in (b) and (c), compute$S[P]$as a function of$\lambda_{1}, \lambda_{2}$(e)$[6$points$]$Setting$\lambda_{1}=\lambda_{2}=\lambda$, discuss what happens to the MaxEnt distribution$P(\sigma)$in the cases$\lambda \rightarrow-\infty, \lambda \rightarrow+\infty$and$\lambda \rightarrow 0$, relating your answer to the value taken by$S[P]$in each of these limits. Part B 问题 4. Consider a random variable$X$taking values in the sample space$\Omega_{X}={0,1,2,3, \ldots}$. Denote by$p_{j}=\operatorname{Prob}[X=j]$with$j \in \Omega_{X} .$Let$Y$be a Poisson random variable with parameter$\lambda$, defined over the same sample space$\Omega_{X}$Consider a system of$N$voters$\sigma_{i} \in{-1,0,1} .$Denote the voting pattern by$\sigma=\left(\sigma_{1}, \ldots, \sigma_{N}\right)$, and assume that the distribution of patterns is $$P(\boldsymbol{\sigma})=\exp (h M(\boldsymbol{\sigma})) / Z \text { , }$$ where$M(\sigma)=\sum_{i=1}^{N} \sigma_{i}$is the magnetisation,$h$the associated Lagrange multiplier, and$Z$the partition function. Define the number of negative votes as$M_{-}(\boldsymbol{\sigma})=\frac{1}{2} \sum_{i=1}^{N}\left(\sigma_{i}^{2}-\sigma_{i}\right)$. In the following, you may use without proof: • the binomial expansion formula$(x+y)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l}n \ k\end{array}\right) x^{k} y^{n-k}$, for$n \in \mathbb{N}$. • the symmetry of binomial coefficients$\left(\begin{array}{c}n \ k\end{array}\right)=\left(\begin{array}{c}n \ n-k\end{array}\right)$, for$0 \leq k \leq n$. • the identity$\sum_{k=0}^{n} k a^{k}\left(\begin{array}{l}n \ k\end{array}\right)=a n(a+1)^{n-1}$. (a) [4 points] Compute the partition function$Z$and the free energy$F$as functions of$h$and$N$. (b) [ 17 points] Using the integral representation of the Kronecker delta $$\delta_{a, b}=\int_{0}^{2 \pi} \frac{\mathrm{d} \xi}{2 \pi} e^{\mathrm{i} \xi(a-b)}$$ for$a, b$integers, compute the probability distribution of the number of negative votes $$P\left(M_{-}\right)=\operatorname{Prob}\left[M_{-}(\boldsymbol{\sigma})=M_{-}\right]=\sum_{\sigma} P(\boldsymbol{\sigma}) \delta_{M_{-}, M_{-}(\sigma)}$$ and check that it is correctly normalised on the alphabet$M_{-} \in{0,1, \ldots, N}$. (c)$[6$points$]$Compute the average number of negative votes $$\left\langle M_{-}(\sigma)\right\rangle=\sum_{M_{-}=0}^{N} P\left(M_{-}\right) M_{-}$$ (d) [3 points] Compute the limits$h \rightarrow 0$and$h \rightarrow \pm \infty$of$\left\langle M_{-}(\boldsymbol{\sigma})\right\rangle$from (c), and provide an intuitive explanation of the results. 问题 5. Consider a one-dimensional Ising chain with$N$spins taking values$\sigma_{i} \in{-1,1}$. The probability of a configuration$\sigma=\left(\sigma_{1}, \ldots, \sigma_{N}\right)$is given by the Boltzmann weight $$P(\sigma)=\exp (-\beta H(\sigma)) / Z$$ where$\beta$is the inverse temperature and$Z$is the partition function. The total energy$H(\sigma)$is given by $$H(\sigma)=-J \sum_{i=1}^{N} \sigma_{i} \sigma_{i+1}-h \sum_{i=1}^{N} \sigma_{i}$$ where$h \in \mathbb{R}$is the external field and$J>0$is the coupling constant. We assume periodic boundary conditions, so$\sigma_{N+1} \equiv \sigma_{1}$. The partition function$Z$can be written as$Z=\lambda_{1}^{N}+\lambda_{2}^{N}$, where $$\lambda_{1,2}=e^{\beta J} \cosh (\beta h) \pm \sqrt{e^{2 \beta J} \sinh ^{2}(\beta h)+e^{-2 \beta J}}$$ are the eigenvalues of the transfer matrix. (a) [3 points] Compute the free energy per spin$f=\lim {N \rightarrow \infty} F / N$, where$F=-(1 / \beta) \ln Z$(b) [ 6 points] Show that the average total energy can be written as$\langle H(\sigma)\rangle=\frac{\partial}{\partial \beta}(\beta F)$(c) [14 points] Show that $$s(\beta, h, J)=\lim {N \rightarrow \infty} \frac{S[P]}{N}=-\frac{\beta}{\lambda_{1}} \frac{\partial}{\partial \beta} \lambda_{1}+\ln \lambda_{1}$$ where$S[P]$is the entropy in natural log units of$P(\sigma)$, and$\lambda_{1}$is the largest of the two eigenvalues of the transfer matrix. You may use without proof the relation between$F$and$S[P]$. (d) [ 7 points] Find the limits of$s(\beta, 0,1)$for$\beta \rightarrow 0$and$\beta \rightarrow \infty$. What happens to$P(\sigma)\$ in these extreme cases? Discuss whether the system undergoes a phase transition, and if it does, determine at which temperature.

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