Thermodynamics of Creation. Note Quote.

Just like the early-time cosmic acceleration associated with inflation, a negative pressure can be seen as a possible driving mechanism for the late-time accelerated expansion of the Universe as well. One of the earliest alternatives that could provide a mechanism producing such accelerating phase of the Universe is through a negative pressure produced by viscous or particle production effects. The viscous pressure contributions can be seen as small nonequilibrium contributions for the energy-momentum tensor for nonideal fluids.

Let us posit the thermodynamics of matter creation for a single fluid. To describe the thermodynamic states of a relativistic simple fluid we use the following macroscopic variables: the energy-momentum tensor T^αβ ; the particle flux vector N^α; and the entropy flux vector s^α. The energy-momentum tensor satisfies the conservation law, T^αβ_;β = 0, and here we consider situations in which it has the perfect-fluid form

T^αβ = (ρ+P)u^αu^β − P g^αβ

In the above equation ρ is the energy density, P is the isotropic dynamical pressure, g^αβ is the metric tensor and u^α is the fluid four-velocity (with normalization u^αu_α = 1).

The dynamical pressure P is decomposed as

P = p + Π

where p is the equilibrium (thermostatic) pressure and Π is a term present in scalar dissipative processes. Usually, it is associated with the so-called bulk pressure. In the cosmological context, besides this meaning, Π can also be relevant when particle number is not conserved. In this case, Π ≡ p_c is called the “creation pressure”. The bulk pressure,  can be seen as a correction to the thermostatic pressure when near to equilibrium, thus, it should be always smaller than the thermostatic pressure, |Π| < p. This restriction, however, does not apply for the creation pressure. So, when we have matter creation, the total pressure P may become negative and, in principle, drive an accelerated expansion.

The particle flux vector is assumed to have the following form

N^α = nu^α

where n is the particle number density. N^α satisfies the balance equation N^α_;α = nΓ, where Γ is the particle production rate. If Γ > 0, we have particle creation, particle destruction occurs when Γ < 0 and if Γ = 0 particle number is conserved.

The entropy flux vector is given by

s^α = nσu^α

where σ is the specific (per particle) entropy. Note that the entropy must satisfy the second law of thermodynamics s^α_;α ≥ 0. Here we consider adiabatic matter creation, that is, we analyze situations in which σ is constant. With this condition, by using the Gibbs relation, it follows that the creation pressure is related to Γ by

p_c = − (ρ+p)/3H Γ

where H = a ̇/a is the Hubble parameter, a is the scale factor of the Friedmann-Robertson-Walker (FRW) metric and the overdot means differentiation with respect to the cosmic time. If σ is constant, the second law of thermodynamics implies that Γ ≥ 0 and, as a consequence, particle destruction (Γ < 0) is thermodynamically forbidden. Since Γ ≥ 0, it follows that, in an expanding universe (H > 0), the creation pressure p_c cannot be positive.

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Dissipations – Bifurcations – Synchronicities. Thought of the Day 29.0

Deleuze’s thinking expounds on Bergson’s adaptation of multiplicities in step with the catastrophe theory, chaos theory, dissipative systems theory, and quantum theory of his era. For Bergson, hybrid scientific/philosophical methodologies were not viable. He advocated tandem explorations, the two “halves” of the Absolute “to which science and metaphysics correspond” as a way to conceive the relations of parallel domains. The distinctive creative processes of these disciplines remain irreconcilable differences-in-kind, commonly manifesting in lived experience. Bergson: Science is abstract, philosophy is concrete. Deleuze and Guattari: Science thinks the function, philosophy the concept. Bergson’s Intuition is a method of division. It differentiates tendencies, forces. Division bifurcates. Bifurcations are integral to contingency and difference in systems logic.

The branching of a solution into multiple solutions as a system is varied. This bifurcating principle is also known as contingency. Bifurcations mark a point or an event at which a system divides into two alternative behaviours. Each trajectory is possible. The line of flight actually followed is often indeterminate. This is the site of a contingency, were it a positionable “thing.” It is at once a unity, a dualism and a multiplicity:

Bifurcations are the manifestation of an intrinsic differentiation between parts of the system itself and the system and its environment. […] The temporal description of such systems involves both deterministic processes (between bifurcations) and probabilistic processes (in the choice of branches). There is also a historical dimension involved […] Once we have dissipative structures we can speak of self-organisation.

Figure: In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. The illustration above shows bifurcations (occurring at the location of the blue lines) of the logistic map as the parameter r is varied. Bifurcations come in four basic varieties: flip bifurcation, fold bifurcation, pitchfork bifurcation, and transcritical bifurcation. 

A bifurcation, according to Prigogine and Stengers, exhibits determinacy and choice. It pertains to critical points, to singular intensities and their division into multiplicities. The scientific term, bifurcation, can be substituted for differentiation when exploring processes of thought or as Massumi explains affect:

Affect and intensity […] is akin to what is called a critical point, or bifurcation point, or singular point, in chaos theory and the theory of dissipative structures. This is the turning point at which a physical system paradoxically embodies multiple and normally mutually exclusive potentials… 

The endless bifurcating division of progressive iterations, the making of multiplicities by continually differentiating binaries, by multiplying divisions of dualities – this is the ontological method of Bergson and Deleuze after him. Bifurcations diagram multiplicities, from monisms to dualisms, from differentiation to differenciation, creatively progressing. Manuel Delanda offers this account, which describes the additional technicality of control parameters, analogous to higher-level computer technologies that enable dynamic interaction. These protocols and variable control parameters are later discussed in detail in terms of media objects in the metaphorical state space of an in situ technology:

[…] for the purpose of defining an entity to replace essences, the aspect of state space that mattered was its singularities. One singularity (or set of singularities) may undergo a symmetry-breaking transition and be converted into another one. These transitions are called bifurcations and may be studied by adding to a particular state space one or more ‘control knobs’ (technically control parameters) which determine the strength of external shocks or perturbations to which the system being modeled may be subject.

Another useful example of bifurcation with respect to research in the neurological and cognitive sciences is Francesco Varela’s theory of the emergence of microidentities and microworlds. The ready-for-action neuronal clusters that produce microindentities, from moment to moment, are what he calls bifurcating “break- downs”. These critical events in which a path or microidentity is chosen are, by implication, creative:

On the basis of this fast dynamic, as in an evolutionary process, one neuronal ensemble (one cognitive subnetwork) finally becomes more prevalent and becomes the behavioral mode for the next cognitive moment. By “becomes more prevalent” I do not mean to say that this is a process of optimization: it resembles more a bifurcation or symmetry-breaking form of chaotic dynamics.

Varela’s “breakdown” is an event triggering diachronic emergence – a timeless interval of reserve potential that “restructures the virtual” (Protevi). Varela, with his deeply phenomenological leanings, and Deleuze and Guattari, with their nomadic surfaces, share a jagged cartography of concurrence. Their leanings vacillate (resonate) between difference and resemblance.