The Ninth International Conference on Guided Self-Organisation (GSO-2018) : Information Geometry and Statistical Physics
Abstracts of the Talks
Souriau-Fisher Metric and Higher Order Extension
We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We conclude with Higher order extension of Souriau model based on works of R..S Ingarden and W. Jaworski. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics.
Imperial College London, United Kingdom
Information loss in phase transitions
There are universal features associated with a range of critical phenomena, such as thermal phase transitions or transitions associated with 'exceptional points' where eigenvectors of a matrix coalesce. These include, for example, the breakdown of adiabatic approximations, the curvature divergence of the parameter-space at critical points, or the loss of information concerning the initial state of the system as one passes through critical points. This talk will sketch some of these ideas that suggest that perhaps exceptional point physics is not all that different from critical phenomena in thermal physics.
University of Albany, USA
Entropic Dynamics: from Information Geometry to Quantum Geometry
Entropic Dynamics (ED) is a framework in which dynamical laws are derived as an application of entropic methods of inference. The dynamics of the probability distribution is driven by entropy subject to constraints that are eventually codified into the phase of a wave function. The central challenge is to identify the relevant physical constraints and, in particular, to specify how those constraints are themselves updated.
In this talk I describe how the information geometry of the space of probabilities is extended to the ensemble-phase space of probabilities and phases. The result is a highly symmetric Riemannian geometry that incorporates the symplectic and complex structures that characterize the geometry of quantum mechanics. The ED that preserves these structures is a Hamiltonian flow and the simplest Hamiltonian suggested by the extended metric leads to quantum mechanics. Thus, in the entropic dynamics framework, Hamiltonians and complex wave functions arise as the natural consequence of information geometry.
Universidad Nacional Autonoma de Mexico (UNAM), Mexico
Self-organization conducted by the dynamics towards the attractor at the onset of chaos
We construct an all-inclusive statistical-mechanical model for self-organization based on the hierarchical properties of the nonlinear dynamics towards the attractors that define the period-doubling route to chaos [1-3]. The aforementioned dynamics imprints a sequential assemblage of the model that privileges progressively lower entropies, while a new set of configurations emerges due to the collective partitioning of the original system into secluded portions. The initial canonical balance between numbers of configurations and Boltzmann-Gibbs (BG) statistical weights is drastically altered and ultimately eliminated by the sequential actions of the attractor. However the emerging set of configurations implies a different and novel entropy growth process that eventually upsets the original loss and has the capability of locking the system into a self-organized state with characteristics of criticality, therefore reminiscent in spirit to the so-called self-organized criticality [4,5].
Some specifics of the approach we develop are: We systematically eliminate access to configurations of an otherwise elementary thermal system model by progressively partitioning it into isolated portions until only remains a subset of configurations of vanishing measure. Each isolated portion becomes essentially a micro-canonical ensemble. The thermal system consists of a large number of (effective) degrees of freedom, each occupying entropy levels with the form of inverse powers of two. The sequential process replaces the original configurations by an emerging discrete scale invariant set of ensemble configurations with allowed entropies that are necessarily inverse powers of two. In doing this we achieve the following results:
1) The constrained thermal system becomes a close analogue of the dynamics towards the multifractal attractor at the period-doubling onset of chaos.
2) The statistical-mechanical properties of the thermal system depart from those of the ordinary Boltzmann-Gibbs form and acquire features from q-statistics.
3) Redefinition of entropy levels as logarithms of the original ones recovers the BG scheme and the free energy Legendre transform property.
Furthermore, the sequences of actions on the entropies associated with the degrees of freedom have the following consequences:
i) Confine degrees of freedom on very few configurations of ever decreasing entropies.
ii) The reduction in numbers of configurations goes down from the initial exponential of the number of degrees of freedom to only one per micro-canonical ensemble, there being an equivalent exponential number of such ensembles for the entire system. As the thermodynamic limit is approached a set of initial configurations with nonzero measure reduces to a set of vanishing measure. But in the process a new set of numbers of configurations develops. These are given by the degeneracies of the micro-canonical ensembles.
iii) The new emerging numbers of configurations grow more slowly than exponentially with the size of the system as they are binomial coefficients.
As with any partition function, the sum of the numbers of configurations times their probabilities cannot vanish nor diverge but be unity. Initially (the BG case) the numbers grow exponentially and the weights also decrease exponentially with system size. After the actions of the attractor the numbers of new ensemble degeneracies grow slower than exponentially and the new weights must now decrease accordingly. The new “canonical” partition function acquires q-exponential weights typical of q-statistics. The precise value of the tuning parameter q and its relationship with the inverse temperature is determined.
 Robledo A., Moyano, L.G., “q-deformed statistical-mechanical property in the dynamics of trajectories en route to the Feigenbaum attractor”, Physical Review E 77, 032613 (2008).
 Robledo, A., “A dynamical model for hierarchy and modular organization: The trajectories en route to the attractor at the transition to chaos”, Journal of Physics: Conf. Ser. 394, 012007 (2012).
 Diaz-Ruelas, A., Robledo, A., “Emergent statistical-mechanical structure in the dynamics along the period-doubling route to chaos”, Europhysics Letters 105, 40004 (2014).
 Bak, P., “How Nature Works” (Copernicus, New York, 1996).
 Watkins, N.W., Pruessner, G., Chapman, S.C., Crosby, N.B., Jensen, J.K., “25 Years of Self-organized Criticality: Concepts and Controversies”, Space Science Reviews 198, 3 (2016).
University of Michigan, USA
Connecting Information Geometry with Geometric Mechanics
We connect information geometry to geometric mechanics in an attempt to provide a unified framework for mechanics and information. It is well-known in that, since the Hamiltonian dynamics is well captured by the symplectic flow in the cotangent bundle T*M, discretizing and representing the dynamics in the product manifold MxM preserves all dynamical information about the mechanical flow. We then compare the symplectic structure on MxM as induced by a divergence function in information geometry with the canonical symplectic structure on T*M that supports the Hamiltonian flow, and establish that any divergence function inducing a statistical structure can also serves as a generating function for the symplectic structure. Note that our identification of information geometry and geometric mechanics is via pull-back of canonical symplectic form from T*M to MxM via a symplectomorphism specified by a divergence function; we have not used dynamics and geometric structures on the TM side. In fact, we propose to decouple the Lagrange dynamics (on the TM side) and the Hamiltonian dynamics (on the T*M side) for information systems, in distinction from their mandatory Legendre coupling in mechanical systems. (Joint work with Melvin Leok.)
Date and Location
March 26 - 28, 2018
Max Planck Institute for Mathematics in the Sciences
see travel instructions
MPI for Mathematics in the Sciences
University of Sydney (Australia)
- Nihat Ay, MPI for Mathematics in the Sciences, Leipzig (Germany)
- Domenico Felice, MPI for Mathematics in the Sciences, Leipzig (Germany)
- Carlos Gershenson, Universidad Nacional Autónoma de México, Computer Sciences Department, Mexico City (Mexico)
- Paolo Gibilisco, Università degli Studi di Roma "Tor Vergata", Facoltà di Economia, Roma (Italy)
- Daniel Polani, University of Hertfordshire, Department of Computer Science, Hatfield (United Kingdom)
- Mikhail Prokopenko, University of Sydney, Sydney (Australia)
- Richard Spinney, University of Sydney, Sydney (Australia)
- Justin Werfel, Harvard University, Cambridge (USA)
- Larry Yaeger, Google Inc., San Francisco (USA)
- G. Çiğdem Yalçın, İstanbul Üniversitesi, İstanbul (Turkey)
Administrative ContactAntje Vandenberg
MPI for Mathematics in the Sciences
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