Abrupt changes of regime appear in many domains, from thermodynamics to living systems. They can present comparable signatures, such as hysteresis, scaling laws, avalanches or bifurcations. These resemblances allow a structured comparison without implying that the mechanisms are identical.
1. A common grammar of shifts
- Control parameter: temperature T, pressure P, density ρ, field B or E, stress σ, flux, error rate, connectivity, noise
- Order variable: fraction of a phase, magnetisation, rigidity, global connectivity, coherence, throughput, viability
- Mechanism: symmetry breaking, competition of minima, barriers, nucleation, feedbacks, dissipation, topology
- Signature: jump, divergence, hysteresis, scaling, absorbing state, topological defects
Universality, critical exponents, renormalisation
In the neighbourhood of certain thresholds, observables follow scaling laws characterised by critical exponents. Microscopically different systems can share the same exponents when they belong to the same universality class, which is formalised by the renormalisation group (Wilson, 1975).
Mean field and the role of fluctuations
Landau-type and mean-field approaches provide a structuring understanding of transitions linked to the appearance of an order and to symmetry breaking. Their main limit appears when fluctuations dominate, notably near critical points and in low dimension.
2. Equilibrium transitions
First-order transitions: discontinuity of thermodynamic quantities, latent heat, coexistence of phases. The kinetics is governed by nucleation and growth, which explains hysteresis effects.
Continuous transitions: no latent heat, emergence of long-range correlations. Susceptibilities can diverge, the correlation length increases strongly, and scaling laws appear.
Dynamic criticality: even when the static transition is well identified, the dynamics near the threshold has universality classes of its own. Characteristic times often increase strongly, which corresponds to critical slowing down.
3. Far from equilibrium: bifurcations and instabilities
Far from equilibrium, phase diagrams often take the form of stability diagrams that delimit the regions of existence of stationary, oscillating or spatially structured states. Reaction-diffusion systems and pattern formation provide classic examples.
In some stochastic non-equilibrium systems, a Lyapunov-type function or an effective potential allows a landscape representation of stability. This possibility establishes a limited bridge with equilibrium transitions.
4. Connectivity, cascades and absorbing states
Percolation
Percolation formalises the appearance of a global connectivity beyond a threshold density of sites or links. It provides a minimal model of a connectivity transition, with scaling laws and universality (Broadbent and Hammersley, 1957).
Transitions to absorbing states
Many non-equilibrium systems present a transition between an active regime and an absorbing regime, where the dynamics can no longer "restart" spontaneously. Population extinction, directed percolation and certain reaction-diffusion processes belong to this family (Hinrichsen, 2000).
5. Rheology, yield stress and jamming
In soft materials, concentrated suspensions, gels and certain amorphous solids, flow appears beyond a threshold of stress or shear rate. Hysteresis, strain localisation and shear bands are frequent.
The jamming perspective links density, effective agitation and stress to the appearance of a jammed state (Liu and Nagel, 1998). Friction deeply modifies the jamming boundary.
6. Quantum transitions and topological phases
At temperatures close to zero, changes of field, pressure, doping or interaction can modify the ground state. Quantum fluctuations play the central role (Sachdev, 2011).
Some transitions are not expressed by a conventional symmetry breaking, but by topological invariants and protected excitations (Hasan and Kane, 2010).
7. Active systems, living systems and social dynamics
Systems of active particles present transitions between disordered motion and ordered collective motion, driven by density and noise. The Vicsek model introduces a flocking transition (Vicsek et al., 1995).
In animal groups, certain measures of long-range correlations have been discussed as signatures of proximity to criticality (Cavagna et al., 2010). In social systems, threshold models describe adoption and cascades when the proportion of adopters exceeds a critical value (Granovetter, 1978).
8. Comparing equilibrium and non-equilibrium
At equilibrium, transitions can be understood through the structure of a thermodynamic potential, the competition between minima and symmetry breaking. Far from equilibrium, shifts are most often read as bifurcations of attractors, instabilities maintained by flows, or transitions to absorbing states.
Stability, fluctuations and scaling structures provide a comparative vocabulary. They do not on their own constitute a unified theory of transitions.
Conclusion and applied stakes
The recurrence of certain threshold dynamics invites us to compare the feedbacks, the constraints, the dissipation and the topology proper to each system. A resemblance of signature is not enough to establish a common mechanism.
This mapping can guide the detection of proximity to thresholds and the design of strategies intended to reduce cascades or to preserve capacities for recovery. Its use requires variables and criteria adapted to each domain.
References
Broadbent, S. R., Hammersley, J. M. (1957). Percolation processes. Mathematical Proceedings of the Cambridge Philosophical Society, 53(3), 629-641.
Hinrichsen, H. (2000). Non-equilibrium critical phenomena and phase transitions into absorbing states. Advances in Physics, 49(7), 815-958.
Liu, A. J., Nagel, S. R. (1998). Jamming is not just cool any more. Nature, 396, 21-22.
Hasan, M. Z., Kane, C. L. (2010). Colloquium: Topological insulators. Reviews of Modern Physics, 82, 3045-3067.
Sachdev, S. (2011). Quantum Phase Transitions (2nd ed.). Cambridge University Press.
Vicsek, T., et al. (1995). Novel type of phase transition in a system of self-driven particles. Physical Review Letters, 75, 1226-1229.
Cavagna, A., et al. (2010). Scale-free correlations in starling flocks. PNAS, 107(26), 11865-11870.
Granovetter, M. (1978). Threshold models of collective behavior. American Journal of Sociology, 83(6), 1420-1443.
Wilson, K. G. (1975). The renormalization group. Reviews of Modern Physics, 47, 773-840.