The description of first-order phase transitions using the linear renormalization group is illustrated with the two-dimensional Ising model. Different linear renormalization groups should be used to describe the system in the neighborhood of criticality and away from it. In the latter case, the renormalization-group trajectories are lines of constant spin expectation value, which map a system below critical temperature in a small magnetic field to a system of free spins at a finite magnetic field. A finite-lattice approximation is also presented. The equation of state and several thermodynamic quantities are obtained; a smooth matching of the behavior near and away from criticality occurs naturally.
The renormalization group for the statistical mechanics of a wave field is described. The hypothesis of scaling of correlations near an order-disorder phase-transition point is written in renormalization-group terms. The renormalization-group equations near a fixed point are investigated. The scaling dimensions (critical indices) of the principal quantities are found in the ∊-approximation. The stability of manycomponent systems and the question of asymptotic symmetry are investigated. The renormalization method is applied to the study of the dynamics of systems near a phase-transition point.
The renormalization-group method is applied to the analysis of phase transitions in systems where the order parameter
s is coupled to a nonordering additional variable
y. A variety of critical and tricritical behaviors or first-order transitions is found as a function of the physical variables and possible macroscopic constraints imposed on the system. For
s2y coupling, the correlation function of
y was found to be governed by a correlation length which is proportional to that of the order parameter, and by a critical index ${\ensuremath{\eta}}_{y}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\eta}}_{y}=2\ensuremath{-}2\ensuremath{\alpha}$; $\ensuremath{\alpha}$ here is the specific-heat exponent of the appropriate unconstrained system. The singular part of the susceptibility, ${\ensuremath{\chi}}_{y}$, has a critical exponent equal to $\ensuremath{\alpha}$, the true specific-heat exponent. When the coupling is
s2y2, a weaker singularity of ${\ensuremath{\chi}}_{y}$ appears. The crossover between this behavior and the one typical to
s2y coupling is calculated.
〈y〉 has a singular part with an exponent $1\ensuremath{-}\ensuremath{\alpha}$, in the unconstrained case. The breakdown of the scaling law related to the correlation function of
y in the constrained case is discussed.
Scaling behavior for first-order phase transitions can be derived alternatively but consistently from renormalization-group, phenomenological, or finite-size considerations. A general analysis of densities at a renormalization-group fixed point demonstrates that if the coexistence of
p distinct phases is possible, then
p distinct eigenvalue exponents must equal the spatial dimensionality. This basic eigenvalue (or scaling) exponent condition can also be derived phenomenologically by various arguments not depending on detailed renormalization-group considerations. A scaling description of first-order phase transitions is presented and extended to finite systems with linear dimensions
L, leading to a rounding proportional to ${L}^{\ensuremath{-}d}$, response-function maxima varying as ${L}^{d}\ensuremath{\propto}N$, and boundary-condition-dependent shifts which may be as large as $\ensuremath{\sim}{L}^{\ensuremath{-}1}$.
If we study the history of science we see produced two phenomena which are, so to speak, each the inverse of the other. Sometimes it is simplicity which is hidden under what is apparently complex; sometimes, on the contrary, it is simplicity which is apparent, and which conceals extremely complex realities. Henri Poincaré Devoting only a short chapter to the renormalization group (RG) is a challenge owing to the wealth of both fundamental concepts and computational tools associated with this term. The RG is indeed encountered in many different domains of theoretical physics, ranging from quantum electrodynamics to second-order phase transitions to fractal growth and diffusion processes – one should rather speak of renormalization group S ! We refer to Brown (1993) and Fisher (1998) for an historical account, to Goldenfeld (1992) and Lesne (1998) and references therein for an overview of the domains of application and variants of RGs. We here emphasize that the RG is a way, maybe the most successful and with no doubt the most systematic and constructive, to derive effective low-dimensional descriptions that capture large-scale and/or long-time behavior . The RG can be extended far beyond the specific scope of critical phenomena, to an iterated multiscale approach allowing the construction of robust and minimal macroscopic models describing the universal large-scale features and asymptotics of a complex system. This generalized viewpoint brings out the close logical and even technical connections that bridge, within a unified framework, perturbative RG for singular series expansions, spin-block RG and momentum-shell RG for critical phenomena, RG for the asymptotic analysis of differential and partial differential equations, and probabilistic RG for the derivation of statistical laws and limit theorems.