物理代写|统计力学代写Statistical mechanics代考|Renormalization group equations, Kadanoff scaling

物理代写|统计力学代写Statistical mechanics代考|Renormalization group equations, Kadanoff scaling

In his first paper Wilson established differential equations for $K_L, B_L$ considered as functions of $L$, and showed that their solutions in the critical region are the Kadanoff scaling forms, Eq. (8.21). To do that, he had to treat the scale factor $L$ as a continuous quantity. In the block-spin picture, $L$ is either an integer or is associated with one. ${ }^{44}$ Consider a small change in length scale, $L \rightarrow L(1+\delta)$, $\delta \ll 1$. Assuming $K_L, B_L$ differentiable functions of $L$, for infinitesimal $\delta$,
\begin{aligned} & K_{L(1+\delta)}-K_L \approx \delta L\left(\frac{\mathrm{d} K_L}{\mathrm{~d} L}\right) \equiv \delta u \ & B_{L(1+\delta)}-B_L \approx \delta L\left(\frac{\mathrm{d} B_L}{\mathrm{~d} L}\right) \equiv \delta v, \end{aligned}
with $u \equiv L\left(\mathrm{~d} K_L / \mathrm{d} L\right), v \equiv L\left(\mathrm{~d} B_L / \mathrm{d} L\right)$. Wilson’s key insight is that the functions $u, v$ depend on $L$ only implicitly, through the $L$-dependence of $K_L, B_L, u=u\left(K_L, B_L^2\right), v=v\left(K_L, B_L^2\right){ }^{45} \mathrm{In}$ the Kadanoff construction, by assembling $2^d$ blocks to make a new block, couplings $K_L, B_L$ are mapped to $K_{2 L}, B_{2 L}$, a mapping independent of the absolute length of the initial blocks. In Wilson’s words, “… the Hamiltonian does not know what the size $L$ of the old block was.” By differentiating Eqs. (8.19) and (8.20) with respect to $L$, he showed that, indeed, $u, v$ do not depend explicitly on $L$. Thus, we have differential equations known as the renormalization group equations,
\begin{aligned} \frac{\mathrm{d} K_L}{\mathrm{~d} L} & =\frac{1}{L} u\left(K_L, B_L^2\right) \ \frac{\mathrm{d} B_L}{\mathrm{~d} L} & =\frac{1}{L} B_L v\left(K_L, B_L^2\right), \end{aligned}

物理代写|统计力学代写Statistical mechanics代考|Ginzburg-Landau theory: Spatial inhomogeneities

Wilson sought a model appropriate for the long-range phenomena we have with critical phenomena involving many lattice sites. To do so, he approximated a lattice of spins as a continuous distribution of spins throughout space. He replaced a discrete system of interacting spins $\left{\sigma_i\right}$ with a continuous spin-density field ${ }^{51} S(\boldsymbol{r})$, the spin density at the point located by position vector $r$. A field description is itself a coarse graining, a view of a system from sufficiently large distances that it appears continuous. Consider (again) a block of spins of linear dimension $L$ centered on the point located by $r$. Define a local magnetization density ( $\sigma_i$ here is not necessarily an Ising spin),
$$S_L(\boldsymbol{r}) \equiv \frac{1}{N_L} \sum_{i \in \text { block at } r}\left\langle\sigma_i\right\rangle,$$
where $N_L=(L / a)^d$ is the number of spins per block, with $a$ the lattice constant. We parameterize $S_L(\boldsymbol{r})$ with the block size because $L$ is not uniquely specified. One wants $L$ large enough that $S_L(\boldsymbol{r})$ does not fluctuate wildly as a function of $r$, but yet small enough that we can use the methods of calculus; finding “physically” infinitesimal lengths is a generic problem in constructing macroscopic theories. How is the “blocking” in Eq. (8.113) different from Kadanoff block spins, Eq. (8.17)? $S_L(\boldsymbol{r})$ is defined in terms of averages $\left\langle\sigma_i\right\rangle$, no assertion is made that all spins are aligned, or that scaling is implied. In what follows we drop the coarse-graining block-size $L$ as a parameter-we’ll soon work in $k$-space where we consider only small wave numbers $k<L^{-1} \ll a^{-1}$.

We may think of the field $S(\boldsymbol{r})$ as an inhomogeneous order parameter in the language of Landau theory. The order parameter (Section 7.7) is a thermodynamic quantity such as magnetization that represents the average behavior of the system as a whole. With $S(\boldsymbol{r})$, we have a local magnetization density that allows for spatial inhomogeneities in equilibrium systems. ${ }^{52}$ Ginzburg-Landau theory is a generalization of Landau theory so that it has a more microscopic character. The Landau free energy $F$ (Eq. (7.79)) is an extensive thermodynamic quantity. It can always be written in terms of a density $\mathcal{F} \equiv F / V$, which for bulk systems is a number having no spatial dependence, but for which there is no harm in writing $F=\int \mathrm{d}^d \boldsymbol{r} \mathcal{F}$. If the magnetization density varies spatially, however, and sufficiently slowly, we can infer that its local value $S(\boldsymbol{r})$ (in equilibrium) represents a minimum of the free energy density at that point, implying that $\mathcal{F}$ varies spatially. The Ginzburg-Landau model finds the total free energy through an integration over spatial quantities,
$$F[S]=\int \mathrm{d}^d \boldsymbol{r} \mathcal{F}(S(\boldsymbol{r})) .$$
Equation (8.114) presents us with a variational problem, the reason we referred to the Landau free energy as a functional ${ }^{53}$ in Section 7.7.1: For given $\mathcal{F}$, what is the spatial configuration $S(\boldsymbol{r})$ that minimizes $F$ ? The calculus of variations answers such questions. We want the functional derivative to vanish (Euler-Lagrange equation, (C.18))
$$\frac{\delta \mathcal{F}}{\delta S}=\frac{\partial \mathcal{F}}{\partial S}-\frac{\mathrm{d}}{\mathrm{d} r}\left(\frac{\partial \mathcal{F}}{\partial S^{\prime}}\right)=0,$$
where $S^{\prime} \equiv \mathrm{d} S / \mathrm{d} r$.

统计力学代考

物理代写|统计力学代写Statistical mechanics代考|Renormalization group equations, Kadanoff scaling

$$K_{L(1+\delta)}-K_L \approx \delta L\left(\frac{\mathrm{d} K_L}{\mathrm{~d} L}\right) \equiv \delta u \quad B_{L(1+\delta)}-B_L \approx \delta L\left(\frac{\mathrm{d} B_L}{\mathrm{~d} L}\right) \equiv \delta v$$

$$\frac{\mathrm{d} K_L}{\mathrm{~d} L}=\frac{1}{L} u\left(K_L, B_L^2\right) \frac{\mathrm{d} B_L}{\mathrm{~d} L}=\frac{1}{L} B_L v\left(K_L, B_L^2\right)$$

物理代写|统计力学代写Statistical mechanics代考|Ginzburg-Landau theory: Spatial inhomogeneities

Wilson 寻求一种适用于我们拥有的涉及许多晶格点的临界现象的长程现象的模型。为 此，他将自旋晶格近似为自旋在整个空间的连续分布。他取代了一个离散的相互作用自 旋系统 $\left[\right.$ 左{|sigma_i|右} 具有连续的自旋密度场 ${ }^{51} S(r)$, 位置向量所在点的自旋密度 $r$. 场 描述本身是粗粒度的，是从足够大的距离看系统的视图，它看起来是连续的。（再次) 考虑线性维度的自旋块 $L$ 以位于的点为中心 $r$. 定义局部磁化密度 $\left(\sigma_i\right.$ 这里不一定是伊辛 自旋)，
$$S_L(\boldsymbol{r}) \equiv \frac{1}{N_L} \sum_{i \in \text { block at } r}\left\langle\sigma_i\right\rangle,$$

$$F[S]=\int \mathrm{d}^d \boldsymbol{r} \mathcal{F}(S(\boldsymbol{r}))$$方程 (8.114) 向我们展示了一个变分问题，我们将 Landau 自由能称为泛函的原因 ${ }^{53}$ 在第 7.7.1 节中: 对于给定的 $\mathcal{F}$ ，什么是空间配置 $S(\boldsymbol{r})$ 最小化 $F$ ? 变分法回答了这些问题。我 们希望泛函导数消失（欧拉-拉格朗日方程，(C.18))
$$\frac{\delta \mathcal{F}}{\delta S}=\frac{\partial \mathcal{F}}{\partial S}-\frac{\mathrm{d}}{\mathrm{d} r}\left(\frac{\partial \mathcal{F}}{\partial S^{\prime}}\right)=0$$

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