Emergence and Reduction Combined in Phase Transitions

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.

24 pp, v2: one minor change. Contribution to the Frontiers of Fundamental Physics (FFP 11) Conference Proceedings

References in corpus (3)

• Berezinskii-Kosterlitz-Thouless Crossover in a Trapped Atomic Gas
• More is the Same; Phase Transitions and Mean Field Theories
• Emergence, Reduction and Supervenience: a Varied Landscape

Cited by in corpus (5)

• Less is Different: Emergence and Reduction Reconciled
• Appearing Out of Nowhere: The Emergence of Spacetime in Quantum Gravity
• On emergence in gauge theories at the 't Hooft limit
• Naturalness, Wilsonian Renormalization, and "Fundamental Parameters" in Quantum Field Theory
• Spontaneous Symmetry Breaking in Quantum Systems: Emergence or Reduction?