Abstract
Condensation phenomena arise through a collective behaviour of particles. They are observed in both classical and quantum systems, ranging from the formation of traffic jams in mass transport models to the macroscopic occupation of the energetic ground state in ultra-cold bosonic gases (Bose–Einstein condensation). Recently, it has been shown that a driven and dissipative system of bosons may form multiple condensates. Which states become the condensates has, however, remained elusive thus far. The dynamics of this condensation are described by coupled birth–death processes, which also occur in evolutionary game theory. Here we apply concepts from evolutionary game theory to explain the formation of multiple condensates in such driven-dissipative bosonic systems. We show that the vanishing of relative entropy production determines their selection. The condensation proceeds exponentially fast, but the system never comes to rest. Instead, the occupation numbers of condensates may oscillate, as we demonstrate for a rock–paper–scissors game of condensates.
Highlights
Condensation phenomena arise through a collective behaviour of particles
Dilute gas of bosonic particles is cooled to a temperature near absolute zero, a finite fraction of bosons may condense into the energetic ground state[10,11,12]
The effective dynamics of the bosons become incoherent and are captured on a macroscopic level in terms of the coupled birth–death processes (1) with rates Gi’j 1⁄4 rij(Ni þ 1)Nj (that is all sij 1⁄4 1 in the rates (2))
Summary
Condensation phenomena arise through a collective behaviour of particles They are observed in both classical and quantum systems, ranging from the formation of traffic jams in mass transport models to the macroscopic occupation of the energetic ground state in ultra-cold bosonic gases (Bose–Einstein condensation). Which states become the condensates has, remained elusive far The dynamics of this condensation are described by coupled birth–death processes, which occur in evolutionary game theory. Long-range phase coherence builds up and quantum physics becomes manifest on the macroscopic scale[13,14] In both the classical and the quantum mechanical context, condensation occurs when one or multiple states become macroscopically occupied (they become condensates), whereas the other states become depleted[15,16]. The temporal evolution of the probability distribution P(N, t) is governed by the classical master equation[17,18]: XS À @tPðN; tÞ 1⁄4 Gi jðNi À 1; Nj þ 1ÞPðN À ei þ ej; tÞ i;j1⁄41
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