One-dimensional Janus fluids. Exact solution and mapping from the quenched to the annealed system

Abstract

The equilibrium properties of a Janus fluid confined to a one-dimensional channel are exactly derived. The fluid is made of particles with two faces (active and passive), so that the pair interaction is that of hard spheres, except if the two active faces are in front of each other, in which case the interaction has a square-well attractive tail. Our exact solution refers to quenched systems (i.e., each particle has a fixed face orientation), but we argue by means of statistical–mechanical tools that the results also apply to annealed systems (i.e., each particle can flip its orientation) in the thermodynamic limit. Comparison between theoretical results and Monte Carlo simulations for quenched and annealed systems, respectively, shows an excellent agreement.

Finite-size effects and thermodynamic limit in one-dimensional Janus fluids

“…The quenched quantities in the TL are provided by equations (2.25), (2.28a), and (2.28b). As proved in Appendix B, equation (2.25) is equivalent to (but more compact than) the Gibbs free energy derived in reference [35] from a completely different method. While in reference [35] the thermodynamic results were derived firectly in the TL from the structural correlation functions, here they have been derived by carefully taking the limit N → ∞ from the configuration integral.…”

“…The procedure can be extended in a straightforward way to a biased choice q = 1 2 , which gives rise to x = 1 2 . The naive expectation would be x = q, but preliminary results in reference [35] showed that either 1 2 < x < q or 1 2 > x > q, depending on whether q > 1 2 or q < 1 2 , respectively. One might reasonable wonder whether the property x = q is a finitesize effect that would disappear in the TL.…”

“…On the other hand, it is possible to carry out biased annealed simulations by introducing a parameter q = 1 2 which favors one of the two possible spin orientations (see Appendix D). As observed in reference [35], the average value x 1 ≡ x does not coincide with q, but a natural question arises as to whether or not the inequality x = q is a finite-size artifact.…”

“…Obviously, g ex TL is the excess Gibbs free energy per particle in the TL. That quantity was evaluated by a completely independent route in reference [35] with the result…”

“…The investigation of reference [35] stimulates a few questions: (i) can the exact derivation of the Gibbs free energy in the TL (N → ∞) be extended to quenched and/or annealed finite-N systems? ; (ii) does the quenched↔annealed equivalence break down at finite N?…”