[02-03-2016]

Using complex models, scientists analyze large systems like fluids, social structures, wildlife interactions or the global economy. These have so many units that researchers can only describe them in terms of probability. They keep the exact microscopic details unspecified and assume the system is in any of the equally likely configurations compatible with a handful of known and measurable macroscopic constraints, like the total energy. There are two main trains of thought for doing this. Either the constraints are considered as strictly fixed, or some margin is allowed on their value. We call these approaches microcanonical and canonical, respectively.

For over 100 years, scientists have assumed that for analyzing large-scale systems with so-called short-range interactions, it doesn’t matter which approach you take, as both lead to the same result. Now Leiden physicist Diego Garlaschelli, mathematician Frank den Hollander and their groups have come up with a different scenario, proving that the microcanonical and canonical approaches can be fundamentally different. The examples they considered come from network theory, where the units in a system are nodes in a network, with some constraint either on the total number of links between them or on the number of links per individual node. In the former case the approaches are equivalent as expected, but now it surprisingly turns out that in the latter case they are not, meaning that it actually matters which approach you take to analyze the system.

Besides its theoretical novelty, the result has important practical implications for the choice of approaches in studying global epidemics or financial crises. ‘Let’s say we look at the social interactions in the spread of epidemics,’ says Garlaschelli. ‘We don’t know for every person the exact amount of connections they have with other people. So for some, we need to make an educated guess. In the microcanonical approach, we would pick a fixed amount of connections for each person, and strictly hang on to it. Now if we make one wrong guess for a person, this error would propagate through our model and we’d make a correlated error with another person. But in the canonical approach, we automatically allow for such uncertainties and one wrong guess doesn’t set off a chain reaction of wrong implications. Our new proof implies that you should choose the canonical one in this case.’

The example may sound straightforward, but the research team had to carry out accurate calculations combining statistical physics and discrete mathematics to prove their claim. ‘The discovery is very important at a theoretical level for our understanding of the fundamentals of statistical physics’, Garlaschelli explains. ‘Many results of statistical physics, which underlie much of our understanding of large real-world systems, may no longer hold.’

Breaking of Ensemble Equivalence in Networks, T. Squartini, J. de Mol, F. den Hollander, D. Garlaschelli, Physical Review Letters

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