Kinetic derivation of the hydrodynamic equations for capillary fluids

Abstract

We study the evolution of a two component fluid consisting of "blue" and "red" particles which interact via strong short range (hard core) and weak long range pair potentials. At low temperatures the equilibrium state of the system is one in which there are two coexisting phases. Under suitable choices of space-time scal-ings and system parameters we first obtain (formally) a mesoscopic kinetic Vlasov-Boltzmann equation for the one particle position and velocity distribution functions, appropriate for a description of the phase segregation kinetics in this system. Further scalings then yield Vlasov-Euler and incompressible Vlasov-Navier-Stokes equations. We also obtain, via the usual truncation of the Chapman-Enskog expansion, com-pressible Vlasov-Navier-Stokes equations.

Theory of agent-based market models with controlled levels of greed and anxiety

“…Agents thus behave as 'contrarians'. Here we generalize this rule in the spirit of [11], to incorporate the possibility that whether agents behave as contrarians or trend-followers may depend dynamically on whether they perceive the market state A(t) to be close to or far from its 'natural' value: pia(t+1) = pia(t)−ηR…”

“…with τ = ±1 and A0 > 0. In the second formula, for τ = −1 the agents behave as a trendfollowers (majority game play) when |A| < A0, but switch to contrarians (minority game play) for |A| > A0; this is the model of [11,12]. For τ = 1 the situation is reversed.…”

“…Averaging (7) over the disorder (the strategies) allows us to obtain from (8,9,10,11) expressions for disorder-averaged observables, such as correlation and response functions:…”

“…The rationale is to create more realistic agent behaviour, based on interpreting A as a measure of the price returns in the market. One example was proposed in [11], corresponding to F [A] = ǫA 3 − A, with ǫ > 0. This model, in which agents are trend-follows for |A| < 1/ √ ǫ but contrarians for |A| > 1/ √ ǫ, can not be solved analytically using the standard generating functional analysis route.…”

“…One could argue that when markets are booming, equity prices tend to rise slowly and the trading volume increases (so agents are trend-followers, as in the majority game [10]), until the magnitude |A| of the overall market bid becomes too large, leading to turmoil and contrarian behaviour. This was the view of [11,12]. Alternatively, one could argue that for small |A| the market would be regarded as normal, and agents would seek advantage by contrarian trading, whereas large |A| would be regarded as anomalous, prompting panic-driven trend-following.…”