Emergence and Reduction Combined in Phase Transitions

TL;DR

This paper explores how emergence and reduction are compatible in phase transitions by analyzing the deduction of novel behaviors through limits and finite parameters, with examples from Lee-Yang theory and renormalization group.

Contribution

It develops a framework showing emergence and reduction coexistence in phase transitions, emphasizing the role of limits and finite parameters in physical reality.

Findings

Deduction of novel behavior via limits of an appropriate parameter N.

Finite N behaviors are physically real and exhibit vivid emergence.

Lee-Yang theory illustrates the deduction of phase transition phenomena.

Abstract

In another paper (Butterfield 2011), one of us argued that emergence and reduction are compatible, and presented four examples illustrating both. The main purpose of this paper is to develop this position for the example of phase transitions. We take it that emergence involves behaviour that is novel compared with what is expected: often, what is expected from a theory of the system's microscopic constituents. We take reduction as deduction, aided by appropriate definitions. Then the main idea of our reconciliation of emergence and reduction is that one makes the deduction after taking a limit of an appropriate parameter $N$. Thus our first main claim will be that in some situations, one can deduce a novel behaviour, by taking a limit $N\to\infty$. Our main illustration of this will be Lee-Yang theory. But on the other hand, this does not show that the $N=\infty$ limit is physically real. For our second main claim will be that in such situations, there is a logically weaker, yet still vivid, novel behaviour that occurs before the limit, i.e. for finite $N$. And it is this weaker behaviour which is physically real. Our main illustration of this will be the renormalization group description of cross-over phenomena.