In biology, many organisms or social insects interact through the process of chemotaxis

For biological systems, the number of interacting bacteria or cells is frequently less than a few thousands so that finite N effects and statistical fluctuations are important. In view of these remarks, it is highly desirable to obtain stochastic kinetic equations that take into account fluctuations and that go beyond the deterministic mean field Keller-Segel model

We have derived generalized Keller-Segel models of chemotaxis taking into account fluctuations. This leads to stochastic kinetic equations instead of deterministic equations

Fluctuations become important close to a critical point, so it is valuable to have a model of chemotaxis going beyond the mean field approximation and taking into account fluctuations

The divergence of the spatial correlation function close to the critical point has been analyzed in detail in previous research for Brownian particles interacting through a binary potential

These particles are described by a stochastic Smoluchowski equation coupled to the markovian field equation

Fluctuations can trigger dynamical phase transitions from one state to the other

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