Swift-Hohenberg equation
| Jack B. Swift and Pierre C. Hohenberg (2008), Scholarpedia, 3(9):6395. | doi:10.4249/scholarpedia.6395 | revision #91841 [link to/cite this article] |
The Swift-Hohenberg equation (SH equation) is a partial differential equation for a scalar field which has been widely used as a model for the study of various issues in pattern formation. These include the effects of noise on bifurcations, pattern selection, spatiotemporal chaos and the dynamics of defects. The SH equation may (stochastic SH equation) or may not (deterministic) include a stochastic noise term. It has been used to model patterns in simple fluids (e.g. Rayleigh-Bénard convection) and in a variety of complex fluids and biological materials, such as neural tissues.
Contents |
Motivation
The motivation for the study which led to the SH equation was the analogy between bifurcations in the hydrodynamic behavior of fluids and the associated pde’s on the one hand, and continuous phase transitions in thermodynamic systems on the other hand.
The specific physical system focused on was Rayleigh-Bénard (RB) convection which had attracted renewed interest among physicists in the early 1970s (Ahlers, 1974).
Classical bifurcation theory is analogous to the mean-field or Landau theory of phase transitions which neglects fluctuation effects. The two theories give the same values for the exponents which describe the scaling of the variables of the theory near the bifurcation or transition point. Classical theory and experiment reveal that the amplitude of the fluid velocity field increases continuously from zero as the square root of the difference of the control parameter and its critical value near the onset of RB convection, and the spontaneous magnetization increases as the square root of the difference of the critical temperature and the temperature, in a Heisenberg ferromagnet according to mean-field theory.
It was natural at that time (1974) to ask whether such fluctuations, whose dramatic effects on continuous thermodynamic transitions had been so successfully elucidated by the scaling and renormalization theories in the period 1965-1975, might also modify the behavior of fluids near hydrodynamic bifurcation points.
Stochastic SH equation
From a formal point of view the effect of fluctuations could be studied by adding a noise term to the deterministic (mean-field) equations of hydrodynamics and averaging the observables over the values of the noise (see Hohenberg and Halperin, 1977).
The only universal noise effect known to exist for the Navier-Stokes equation is the thermal noise arising out of the molecular motion of the constituents. Terms representing this thermal noise had been added to the Navier-Stokes equation by Landau and Lifshitz. For the specific physical system of RB convection, the linear effects of these terms had been studied by Zaitsev and Shliomis (1971) and by Graham and shown to be proportional to the ratio of the thermal energy \(kT\ ,\) divided by a typical energy of a Rayleigh-Benard cell, which can be estimated as the dissipation energy per unit volume \(\rho\nu \kappa/d^2\) times the volume \(d^3\ ,\) i.e. the ratio is of order \(kT/ \rho d\nu \kappa\ .\) Here \(k\) is Boltzmann's constant, \(T\) the absolute temperature, \(\rho\) the density of the convecting fluid, \(\nu\) its kinematic viscosity, \(\kappa\) the thermal diffusivity, and \(d\) is the spacing between the horizontal plates which confine the fluid. This quantity, which represents the ratio of a microscopic energy to a macroscopic energy is of order \(10^{-9}\) for typical systems, but there may be circumstances for which it could be made larger (see below) and in any case the question of the effect of fluctuations on bifurcations is of intrinsic theoretical interest even if their magnitude is extremely small.
In the renormalization group theory of thermodynamic phase transitions the crucial features determining the critical behavior (exponents) are the dimension of space (more precisely the number of dimensions for which the system is effectively infinite) and the symmetry of the ‘order parameter’ of the transition to the ordered state. For RB convection the dimension is two and the symmetry is that of an isotropic liquid undergoing a transition to a solid or a liquid crystal. The present situation, however, is different from the usual breaking of rotation symmetry, which occurs for example in a Heisenberg ferromagnet where the order-parameter fluctuations involve small wavevectors coalescing around a definite wavevector. Here there is no underlying fixed lattice so only the magnitude \(q_o\) of the wavevector is fixed and fluctuations occur about a shell in wavevector space, with radius \(q_o\ .\)
The simplest model obeying the requisite symmetries is one with a Lyapunov or free-energy functional of the form (Swift and Hohenberg, 1977) \[ \Phi =- \int d \vec{x} \lbrace \frac{1}{2}\epsilon \psi^2 -\frac{1}{4} \tilde{g}_3\psi^4 -\frac{1}{2} \tilde{\xi}_0^4 \left[ (\nabla^2 + q_0^2)\psi \right]^2 \rbrace \ ,\] from which one can deduce the equation of motion \[ \tau_0 \partial_t \psi = - \frac{\delta \Phi}{\delta \psi} + f(\vec{x},t) = \left[ \epsilon - \tilde{\xi}_0^4 (\nabla^2 + q_0^2)^2 - \tilde{g}_3 \psi^2 \right] \psi(\vec{x},t) + f(\vec{x},t) \ ,\] with \[ \langle f(\vec{x},t) f(\vec{x'},t') \rangle =2F \tau_0 \delta(t-t') \tilde{\xi}_0^2 \delta(\vec{x}-\vec{x}') \] by using the methods which had become standard in the study of the dynamics of phase transitions (Hohenberg and Halperin, 1977). It is the third (differential) term in the free energy, leading to the corresponding linear term in the equation of motion, which ensures the unusual, Brazovskii, symmetry of the system and leads to the differences in critical behavior from those of systems whose free energy is of the more standard Ginzburg-Landau form, such as a Heisenberg ferromagnet. For applications to RB convection one may interpret \(\psi\) as proportional to the difference in the local temperature of the fluid and a linear profile between the confining plates and also \(F\)is proportional to the small parameter of order \(10^{-9}\) in ordinary liquids (Hohenberg and Swift, 1992).
Phase transitions of the Brazovskii symmetry class
The type of phase transition associated with the above free energy had already been studied by Brazovskii (1975) who was motivated by the example of antiferromagnetic liquids or liquid crystals. He had shown that fluctuations would have the effect of changing a transition which was of second order in mean-field theory to a first-order (i.e. discontinuous) transition.
A similar argument was then constructed by adapting Brazovskii’s argument to the two-dimensional case, with the conclusion that fluctuations would make the transition in RB convection discontinuous in contrast to the continuous transition predicted by mean-field theory. Of course the effect would only be observable very close to the bifurcation, in view of the smallness of the noise term.
Although it took many years, the prediction of a discontinuous transition in standard RB convection was confirmed experimentally by Ahlers and coworkers (Oh and Ahlers, 2003). These authors worked with a fluid near its thermodynamic critical point where the thermal diffusivity approaches zero, and with a small plate spacing \(d\ .\) Both factors increase the strength of the fluctuations so that the noise parameter \(F\) is orders of magnitude larger than in ordinary fluids. Although there are some remaining discrepancies between the measurements and the theoretical predictions, a fit of the data to the theory yields a magnitude for \(F\) in remarkably good agreement with the value derived from the fluid parameters.
Deterministic SH equation and pattern formation
More importantly, however, once a simplified equation of motion with many of the essential features of RB convection had been presented a crucial next step was taken. This was to forget the origin of the model as a means of studying fluctuation effects and instead to take it seriously as a model for nonlinear pattern formation, i.e. to seek analytic and numerical solutions of the (mean-field) equations without the noise term (\(F = 0\)).
The important initial steps in this direction were taken by Manneville, Cross, Greenside and Coughran, and especially Cross and Greenside. The last authors provided detailed analysis of the patterns above threshold, and also generalized the model by modifying the nonlinear terms in various ways. References to these papers and selected results from them are given in the review of Cross and Hohenberg (1993).
Further developments
The above works and subsequent analyses elucidated those features of RB convection which are retained by the SH model and those which are omitted. For example, the SH equation derives from a potential and thus does not possess persistent chaotic solutions. Various nonpotential modifications were introduced by Cross and Greenside and others and their effects have been studied. A particularly instructive set of simulations of the SH equation has been presented by Cross (SH simulations by Michael Cross).
Generally speaking it is this ‘second life’ of the SH equation as a deterministic model for pattern formation which is responsible for its unusual popularity, arising from the relative ease of simulation and analysis, as well as the many modifications that can be introduced to model specific experimental effects.
As of 2007, the S-H paper has garnered 558 citations, according to the ISI record. The rate of new citations has been 21 in 2005, 23 in 2006 and 22 in 2007, with the highest number, 31, garnered in 2003. Overall the articles cover mostly fluid dynamics and pattern formation in a variety of complex fluids and biological materials such as neural tissues. There are also articles on applications to patterns in block copolymers and active nonlinear optical media, as well as articles of a more theoretical nature, studying problems in dynamical systems and chaos.
References
- G. Ahlers, in Fluctuations, Instabilities, and Phase Transitions, edited by T. Riste (Plenum, NY, 1974), p. 181.
- S. A. Brasovskii, Zh. Eksp. Teor. Fiz. 68, 175 (1975) [Sov. Phys.-JETP 41, 85 (1975)].
- M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
- P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).
- P. C. Hohenberg and J. B. Swift, Phys. Rev. A46, 4773 (1992).
- J. Oh and G. Ahlers, Phys. Rev. Lett. 91, 095401 (2003).
- J. Swift and P. C. Hohenberg, Phys. Rev. A15, 319 (1977).
- V. M. Zaitsev and M. I. Shliomis, Zh. Eksp. Teor. Fiz. 59, 1583 (1970) [Sov. Phys.-JETP 32, 866 (1971)].
Internal references
- Jan A. Sanders (2006) Averaging. Scholarpedia, 1(11):1760.
- John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Giovanni Gallavotti (2008) Fluctuations. Scholarpedia, 3(6):5893.
- Rodolfo Llinas (2008) Neuron. Scholarpedia, 3(8):1490.
- Hermann Haken (2008) Self-organization. Scholarpedia, 3(8):1401.
