Analysis

Since a detailed mathematical analysis of this type of system has already been made[9,10], we will confine ourselves here to a brief statement of the pertinent results.

We analyze the case where the relaxation of the field is very slow in comparison to the relaxation time of the particle density . In this case the field is said to slave[18] to the particle denisty, and the particle density can be removed from the picture in the following way. Since the field changes very little on these time scales we will assume it is constant as the particles equilibrate in the energy landscape . It is easy to show that the equilibrium distribution of particles evolves to

Elimination of the particle variables is made possible by substituting this expression into Eq. 2, so that

where N is the total number of particles, and where we neglect diffusion and will not consider it futher in the present letter. For our present purpose we will only be interested in the stability of the uniform phase of the system, , which we will treat by mean-field theory. The above equation has a mean-field solution of , where .

The stability of this solution is found by expanding about the solution with . This leads to the mean-field stability criterion

This is the most important theoretical result, and it gives the location of the second-order phase transition, that is, the location of the boundary separating totally random behavior from ordered behavior of varying types in the physiological phase space. As shown in [9,10], ordered behavior sets in when this criterion is broken. This criterion is true for any energy function , and allows us to calculate the physiological phase boundaries in every case.

For the particular behavioral energy function used here (Eq. 3) the transition lies along the curve

The symmetry which allows for a mean-field type solution for the location of the transition line case (detailed balance) is maintained up to the point of the transition and is effectively spontaneously broken at that point. This means the points of transition from disorder to order can be determined theoretically, but not the resulting patterns, which must be determined via simulations or further theoretical analysis.

In addition to being the major landmark in the physiological parameter space, we believe both on general grounds, and because of the results presented in the next sections, that the location of this line has significant behavioral implications. Clearly it is important that ordered behavior exists so that ordered patterns of flow can form, but just as importantly, the behavior should not be too rigid and ordered since fluctuation and instabilities might increase the flexibility of response of the mass action. Thus, if the dynamics are such that there can be significant fluctuations in the patterns of mass action, the swarms could better respond to a changing environment. We might conclude that a ``good" place to be would be in the order region, but near to the transition line in such a way as to optimize the conflicting tendencies of controlled order behaviors versus flexible random behavior. We will also see in the next sections that large fluctuation actually serves to stabilize some of the important patterns of collective behavior of the system.