KEYWORDS: Particles, Stochastic processes, Complex systems, Systems modeling, Computer simulations, Monte Carlo methods, Solids, Physics, Homogenization, Process modeling
A free zero-range process (FRZP) is a simple stochastic process describing the dynamics of a gas of particles
hopping between neighboring nodes of a network. We discuss three different cases of increasing complexity: (a)
FZRP on a rigid geometry where the network is fixed during the process, (b) FZRP on a random graph chosen
from a given ensemble of networks, (c) FZRP on a dynamical network whose topology continuously changes
during the process in a way which depends on the current distribution of particles. The case (a) provides a
very simple realization of the phenomenon of condensation which manifests as the appearance of a condensate
of particles on the node with maximal degree. A particularly interesting example is the condensation on scalefree
networks. Here we will model it by introducing a single-site inhomogeneity to a k-regular network. This
simplified situation can be easily treated analytically and, on the other hand, shows quantitatively the same
behavior as in the case of scale-free networks. The case (b) is very interesting since the averaging over typical
ensembles of graphs acts as a kind of homogenization of the system which makes all nodes identical from the point
of view of the FZRP. In effect, the partition function of the steady state becomes invariant with respect to the
permutations of the particle occupation numbers. This type of symmetric systems has been intensively studied
in the literature. In particular, they undergo a phase transition to the condensed phase, which is caused by a
mechanism of spontaneous symmetry breaking. In the case (c), the distribution of particles and the dynamics
of network are coupled to each other. The strength of this coupling depends on the ratio of two time scales:
for changes of the topology and of the FZRP. We will discuss a specific example of that type of interaction and
show that it leads to an interesting phase diagram. The case (b) mentioned above can be viewed as a limiting
case where the typical time scale of topology fluctuations is much larger than that of the FZRP.
Conference Committee Involvement (1)
Noise and Stochastics in Complex Systems and Finance
Access to the requested content is limited to institutions that have purchased or subscribe to SPIE eBooks.
You are receiving this notice because your organization may not have SPIE eBooks access.*
*Shibboleth/Open Athens users─please
sign in
to access your institution's subscriptions.
To obtain this item, you may purchase the complete book in print or electronic format on
SPIE.org.
INSTITUTIONAL Select your institution to access the SPIE Digital Library.
PERSONAL Sign in with your SPIE account to access your personal subscriptions or to use specific features such as save to my library, sign up for alerts, save searches, etc.