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DY: Fachverband Dynamik und Statistische Physik

DY 55: Poster: Noneq. Stat. Phys., Stat. Bio. Phys., Brownian

DY 55.3: Poster

Donnerstag, 4. April 2019, 15:00–18:00, Poster B2

Field Theoretic Thermodynamic Uncertainty Relation for the Kardar-Parisi-Zhang Equation — •Oliver Niggemann and Udo Seifert — II. Institut für Theoretische Physik, Universität Stuttgart

In this poster I will present a way to show the validity of the thermodynamic uncertainty relation in the setting of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation with white in time, spatially colored Gaussian noise for weak coupling, i.e. ∂_th(x,t)=ν∂_x²h(x,t)+λ/2(∂_xh(x,t))²+η(x,t) with ⟨η(x,t)⟩=0 and ⟨η(x,t)η(x^′,t^′)⟩=D(|x−x^′|)δ(t−t^′). Within Hilbert space theory of stochastic partial differential equations the height field h(x,t) is regarded as a trajectory t↦ h(x,t), h(x,t)∈H. On a finite spatial interval [0,L], H=L₂(0,L) is a suitable choice, and the eigenfunctions of the differential operator A≡ν∂_x² with periodic boundary conditions form an orthonormal base of H. The height h(x,t) and the noise η(x,t) are expressed then by means of eigenfunction expansions. In this setting, the following thermodynamic uncertainty relation has been derived for the non-equilibrium steady state: Δ s_tot^stat є²>2+O(λ²). Here Δ s_tot^stat≡∫₀^t dt^′((∂_xh(x,t^′))²,D⁻¹(∂_xh(x,t^′))²)_L2 is the stationary state total entropy production, and є²≡⟨∥ h(x,t)−⟨ h(x,t)⟩∥_L2²⟩/∥⟨ h(x,t)⟩∥_L2² is the squared variation coefficient of h(x,t), evaluated for t≫1. D⁻¹ is the inverse of the integral covariance operator with kernel D(|x−x^′|). The uncertainty relation holds for D(x) describing a class of regular spatially colored noise, which, in a suitable limit, approximates white noise and for white noise itself, i.e. D(x)=δ(x).