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Quantum randomness in the Sky

Sayantan Choudhury, Arkaprava Mukherjee

2019
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European Physical Journal C: Particles and Fields
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In this article, we study quantum randomness of stochastic cosmological particle production scenario using quantum corrected higher order Fokker Planck equation. Using the one to one correspondence between particle production in presence of scatters and electron transport in conduction wire with impurities we compute the quantum corrections of Fokker Planck Equation at different orders. Finally, we estimate Gaussian and non-Gaussian statistical moments to verify our result derived to explain
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... chastic particle production probability distribution profile. It is a well known fact that the particle production scenario in the early universe cosmology (during reheating) follows the dynamical master equation, aka Klein-Gordon equation. On the other hand, transport phenomena of electron through a conduction wire with impurities follow time independent Schrödinger equation. Both of this dynamical time dependent phenomena have structural one to one correspondence [1,2]. Anderson Localization and saturation of the chaos are some well studied phenomena in the context of scattering problem can be extended to describe the quantum randomness phenomena during cosmological particle production. From their inherent stochastic nature quantum chaos can be related to them and chaos bound can be defined either by Lyapunov exponent [3] or by Spectral Form Factor [4, 5] . The possible quantum effects arising from higher order corrections in dynamical master equation aka Fokker Planck equation for particle production scenario in the early universe cosmology (during reheating) can be achieved from the present discussion. For comparing scattering event with stochastic particle production Dirac Delta profile of time dependent coupling (mass function) is chosen, This project is the part of the non-profit virtual international research consortium "Quantum Structures of the Space- localized at time scale τ = τ j (where j represents the number of non-adiabatic events). Further using the concept of transfer matrices occupation number can be computed from this set up. To model a phenomenological situation where width (w j ) of the profile of the time dependent coupling is finite and the scattering event is relevant, we consider sech scatterers. It is important to note that, in the limit w j → ∞ the Dirac Delta profile can be recovered from this phenomenological profile. In the context of dissipative system, Fokker Planck equation explains the probability density for particle position of Brownian motion in a random system. For a Markovian process this situation can be expressed by Chapman-Kolmogorov equation [1] . Now considering Maximum Entropy Anstaz we can derive the Fokker Planck equation from Smoluchowski equation when we integrate the probability density over the angular coordinate θ : P(n, θ, φ; τ + δτ ) ≡ P(n, θ; τ + δτ ) → P(n + δn; τ ) δτ (2) where we consider an infinitesimal change (δθ) is not functionally dependent on θ . Further Taylor expansion of P(n + δn; τ ) δτ with respect to the infinitesimal occupation number (δn) with the constraint in this context P(n; τ ) δτ = P(n, τ ) gives the following result: P(n + δn; τ ) δτ = P(n; τ ) δτ + ∞ q=1 (q!) −1 ∂ q n P(n; τ ) δτ This gives the following general structure of Fokker-Planck equation which we will use for our all calculations: 0123456789().: V,-vol 123

doi:10.1140/epjc/s10052-019-7072-1
fatcat:x4sla7ycbveyxgxmcdn6bumay4