• DOI:10.1007/s10955-012-0481-2
• Corpus ID: 119132309

Non equilibrium stationary state for the SEP with births and deaths

@inproceedings{Masi2012NonES,
  title={Non equilibrium stationary state for the SEP with births and deaths},
  author={Anna de Masi and Errico Presutti and Dimitrios K. Tsagkarogiannis and Maria Eulalia Vares},
  year={2012}
}

• A. Masi, E. Presutti, +1 author M. Vares
• Published 8 March 2012
• Mathematics

This paper is a follow-up of the study initiated in [1], [2], where current reservoirs in the context of stochastic interacting particle systems have been proposed as a method to investigate stationary non-equilibrium states with steady currents produced by action at the boundary. Due to the particular difficulties in implementing this new method, we consider the simplest possible particle system. The bulk dynamics is the symmetric simple exclusion process (SSEP) in the interval ΛN = [−N,N ] ∩ Z (N a positive integer and N → ∞ eventually), namely the state space is {0, 1}ΛN (at most one particle per site): independently each particle tries to jump at rate N/2 to each one of its nearest neighbor (n.n.) sites, the jump then takes place if and only if the chosen site is empty, jumps outside ΛN are suppressed. To induce a current we send in particles from the right and take them out from the left, and would like this to happen at rate Nj/2, j > 0 a fixed parameter independent of N . Due to the restrictions imposed by the configurational space, we have to be more precise when defining this dynamics. For this we fix a parameter K ≥ 1 (an integer) and two intervals I± of length K at the boundaries: I+ ≡ [N −K + 1, N ] and I− ≡ [−N,−N +K − 1]. At rate Nj/2, when I+ is not totally occupied, we create a particle at its rightmost empty site; with the same rate, unless I− is empty, we take out a particle from its leftmost occupied site. In case I+ is already full, or I− empty, the corresponding mechanism aborts. In [1], [2] we have proved that at any time t > 0 propagation of chaos holds and that in the limit N → ∞ the hydrodynamical equation is the linear heat equation: