Particle entity in the Doi–Peliti and response field formalisms

Abstract

We introduce a procedure to test a theory for point particle entity, that is, whether said theory takes into account the discrete nature of the constituents of the system. We then identify the mechanism whereby particle entity is enforced in the context of two field-theoretic frameworks designed to incorporate the particle nature of the degrees of freedom, namely the Doi-Peliti field theory and the response field field theory that derives from Dean’s equation. While the Doi-Peliti field theory encodes the particle nature at a very fundamental level that is easily revealed, demonstrating the same for Dean’s equation is more involved and results in a number of surprising diagrammatic identities. We derive those and discuss their implications. These results are particularly pertinent in the context of active matter, whose surprising and often counterintuitive phenomenology rests wholly on the particle nature of the agents and their degrees of freedom as particles.

Accurate dynamics from self-consistent memory in stochastic chemical reactions with small copy numbers

“…In the continuous time and space limit, the Hamiltonian simply acquires an additional term −D φ∇ 2 ϕ, with D the diffusion coefficient and ϕ, φ now also dependent on position. Doi-Peliti field theory with diffusing particles preserves their discrete particle identities [59], i.e. the field theory maintains the particle nature of the degrees of freedom.…”

Global density equations for a population of actively switching particles

“…The basic idea is to apply a MSRJD path integral construction [66][67][68] to the DK equation obtained by setting A n (x) = 0 in equation (3.11). (For a complementary approach based on a Doi-Peliti path integral formulation [69][70][71], see [72]. Note, however, that care has to be taken when comparing Doi-Peliti and MSRJD field theories since the actual fields have different physical interpretations.)…”

“…Equation (5.19) is the starting point for performing various diagrammatic expansions by Taylor expanding the functional operator e SI . This has been carried out elsewhere for a nonswitching Brownian gas [65,72] whose MSRJD action is of the form…”