English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We give a first principles derivation of the stochastic partial differentialequations that describe the chemical reactions of the Gray-Scott model (GS):$U+2V {\stackrel {\lambda}{\rightarrow}} 3 V;$ and $V {\stackrel{\mu}{\rightarrow}} P$, $U {\stackrel {\nu}{\rightarrow}} Q$, with a constantfeed rate for $U$. We find that the conservation of probability ensured by thechemical master equation leads to a modification of the usual differentialequations for the GS model which now involves two composite fields and alsointrinsic noise terms. One of the composites is $\psi_1 = \phi_v^2$, where $ <\phi_v >_{\eta} = v$ is the concentration of the species $V$ and the averagingis over the internal noise $\eta_{u,v,\psi_1}$. The second composite field isthe product of three fields $ \chi = \lambda \phi_u \phi_v^2$ and requires anoise source to ensure probability conservation. A third composite $\psi_2 =\phi_{u} \phi_{v}$ 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 toparticle number conservation. By expanding around the (broken symmetry) zeroenergy solution of the Hamiltonian (by performing a Doi shift) one obtains fromour path integral formulation the usual reaction diffusion equation, at theclassical level. The Langevin equations that are derived from the chemicalmaster equation have multiplicative noise sources for the density fields$\phi_u, \phi_v, \chi$ that induce higher order processes such as $n\rightarrow n$ scattering for $n > 3$. The amplitude of the noise acting on $\phi_v$ is itself stochastic in nature.

Composite Bound States and Broken U(1) symmetry in the Chemical Master Equation derivation of the Gray-Scott Model