More is the Same; Phase Transitions and Mean Field Theories

This paper looks at the early theory of phase transitions. It considers a group of related concepts derived from condensed matter and statistical physics. The key technical ideas here go under the names of "singularity", "order parameter", "mean field theory", and "variational method". In a less technical vein, the question here is how can matter, ordinary matter, support a diversity of forms. We see this diversity each time we observe ice in contact with liquid water or see water vapor, "steam", come up from a pot of heated water. Different phases can be qualitatively different in that walking on ice is well within human capacity, but walking on liquid water is proverbially forbidden to ordinary humans. These differences have been apparent to humankind for millennia, but only brought within the domain of scientific understanding since the 1880s. A phase transition is a change from one behavior to another. A first order phase transition involves a discontinuous jump in a some statistical variable of the system. The discontinuous property is called the order parameter. Each phase transitions has its own order parameter that range over a tremendous variety of physical properties. These properties include the density of a liquid gas transition, the magnetization in a ferromagnet, the size of a connected cluster in a percolation transition, and a condensate wave function in a superfluid or superconductor. A continuous transition occurs when that jump approaches zero. This note is about statistical mechanics and the development of mean field theory as a basis for a partial understanding of this phenomenon.Comment: 25 pages, 6 figure

Reflections on the four facets of symmetry: how physics exemplifies rational thinking

In contemporary theoretical physics, the powerful notion of symmetry stands for a web of intricate meanings among which I identify four clusters associated with the notion of transformation, comprehension, invariance and projection. While their interrelations are examined closely, these four facets of symmetry are scrutinised one after the other in great detail. This decomposition allows us to examine closely the multiple different roles symmetry plays in many places in physics. Furthermore, some connections with others disciplines like neurobiology, epistemology, cognitive sciences and, not least, philosophy are proposed in an attempt to show that symmetry can be an organising principle also in these fields

Rigorous Results, Cross-Model Justification, and the Transfer of Empirical Warrant: The Case of Many-Body Models in Physics

This paper argues that a successful philosophical analysis of models and simulations must accommodate an account of mathematically rigorous results. Such rigorous results may be thought of as genuinely model-specific contributions, which can neither be deduced from fundamental theory nor inferred from empirical data. Rigorous results provide new indirect ways of assessing the success of models and simulations and are crucial to understanding the connections between different models. This is most obvious in cases where rigorous results map different models on to one another. Not only does this put constraints on the extent to which performance in specific empirical contexts may be regarded as the main touchstone of success in scientific modelling, it also allows for the transfer of warrant across different models. Mathematically rigorous results can thus come to be seen as not only strengthening the cohesion between scientific strategies of modelling and simulation, but also as offering new ways of indirect confirmation

Air pollution Exposure Model for Individuals (EMI) in health studies: Evaluation for ambient PM_2.5 in Central North Carolina.

Air pollution health studies of fine particulate matter (diameter ≤2.5 μm, PM2.5) often use outdoor concentrations as exposure surrogates. Failure to account for variability of indoor infiltration of ambient PM2.5 and time indoors can induce exposure errors. We developed and evaluated an exposure model for individuals (EMI), which predicts five tiers of individual-level exposure metrics for ambient PM2.5 using outdoor concentrations, questionnaires, weather, and time-location information. We linked a mechanistic air exchange rate (AER) model to a mass-balance PM2.5 infiltration model to predict residential AER (Tier 1), infiltration factors (Tier 2), indoor concentrations (Tier 3), personal exposure factors (Tier 4), and personal exposures (Tier 5) for ambient PM2.5. Using cross-validation, individual predictions were compared to 591 daily measurements from 31 homes (Tiers 1-3) and participants (Tiers 4-5) in central North Carolina. Median absolute differences were 39% (0.17 h(-1)) for Tier 1, 18% (0.10) for Tier 2, 20% (2.0 μg/m(3)) for Tier 3, 18% (0.10) for Tier 4, and 20% (1.8 μg/m(3)) for Tier 5. The capability of EMI could help reduce the uncertainty of ambient PM2.5 exposure metrics used in health studies