Gray-Scott模型化学主方程推导中的复合束缚态和U（1）对称性破缺
Composite Bound States and Broken U(1) symmetry in the Chemical Master Equation derivation of the Gray-Scott Model
摘要：
我们给出了描述Gray-Scott模型（GS）的化学反应的随机偏微分方程的第一原理推导：$U+2V {\stackrel λ{\rightarrow}} 3 V;$和$V {\stackrel μ{\rightarrow}} P$，$U {\stackrel ν{\rightarrow}} Q$，对于$U$具有恒定的进料速率。我们发现，由化学主方程确保的概率守恒导致GS模型的通常的微分方程的修改，现在涉及两个复合领域，也内在的噪声项。其中一种复合物是$ψ_1 = ϕ_v^2$，其中$ < ϕ_v >_η = v$是物质$V$的浓度，平均值是内部噪声$η_{u,v,ψ_1}$。第二复合字段是三个字段$ χ= λϕ_u ϕ_v^2$的乘积，并且需要噪声源以确保概率守恒。第三种复合物$ψ_2 = ϕ_{u} ϕ_{v}$也可以从噪声诱导的反应中鉴定出来。支配多体波函数时间演化的哈密顿量，与主方程相关，具有与粒子数守恒相关的破缺U（1）对称性。通过扩展周围的（对称性破缺）零能量解决方案的哈密顿量（通过执行Doi移位）从我们的路径积分公式得到通常的反应扩散方程，在经典水平。从化学主方程导出的朗之万方程具有用于密度场$ϕ_u, ϕ_v, χ$的乘性噪声源，该乘性噪声源诱导更高阶的过程，诸如用于$n > 3$的$n \rightarrow n$散射。作用于$ ϕ_v$的噪声幅度本身在性质上是随机的。
英文摘要：
We give a first principles derivation of the stochastic partial differential equations that describe the chemical reactions of the Gray-Scott model (GS): $U+2V {\stackrel λ{\rightarrow}} 3 V;$ and $V {\stackrel μ{\rightarrow}} P$, $U {\stackrel ν{\rightarrow}} Q$, with a constant feed rate for $U$. We find that the conservation of probability ensured by the chemical master equation leads to a modification of the usual differential equations for the GS model which now involves two composite fields and also intrinsic noise terms. One of the composites is $ψ_1 = ϕ_v^2$, where $ < ϕ_v >_η = v$ is the concentration of the species $V$ and the averaging is over the internal noise $η_{u,v,ψ_1}$. The second composite field is the product of three fields $ χ= λϕ_u ϕ_v^2$ and requires a noise source to ensure probability conservation. A third composite $ψ_2 = ϕ_{u} ϕ_{v}$ can be also be identified from the noise-induced reactions. The Hamiltonian that governs the time evolution of the many-body wave function, associated with the master equation, has a broken U(1) symmetry related to particle number conservation. By expanding around the (broken symmetry) zero energy solution of the Hamiltonian (by performing a Doi shift) one obtains from our path integral formulation the usual reaction diffusion equation, at the classical level. The Langevin equations that are derived from the chemical master equation have multiplicative noise sources for the density fields $ϕ_u, ϕ_v, χ$ that induce higher order processes such as $n \rightarrow n$ scattering for $n > 3$. The amplitude of the noise acting on $ ϕ_v$ is itself stochastic in nature.
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