# Econophysics: Financial White Noise Switch. Thought of the Day 115.0 What is the cause of large market fluctuation? Some economists blame irrationality behind the fat-tail distribution. Some economists observed that social psychology might create market fad and panic, which can be modeled by collective behavior in statistical mechanics. For example, the bi-modular distribution was discovered from empirical data in option prices. One possible mechanism of polarized behavior is collective action studied in physics and social psychology. Sudden regime switch or phase transition may occur between uni-modular and bi-modular distribution when field parameter changes across some threshold. The Ising model in equilibrium statistical mechanics was borrowed to study social psychology. Its phase transition from uni-modular to bi-modular distribution describes statistical features when a stable society turns into a divided society. The problem of the Ising model is that its key parameter, the social temperature, has no operational definition in social system. A better alternative parameter is the intensity of social interaction in collective action.

A difficult issue in business cycle theory is how to explain the recurrent feature of business cycles that is widely observed from macro and financial indexes. The problem is: business cycles are not strictly periodic and not truly random. Their correlations are not short like random walk and have multiple frequencies that changing over time. Therefore, all kinds of math models are tried in business cycle theory, including deterministic, stochastic, linear and nonlinear models. We outline economic models in terms of their base function, including white noise with short correlations, persistent cycles with long correlations, and color chaos model with erratic amplitude and narrow frequency band like biological clock. The steady state of probability distribution function in the Ising Model of Collective Behavior with h = 0 (without central propaganda field). a. Uni-modular distribution with low social stress (k = 0). Moderate stable behavior with weak interaction and high social temperature. b. Marginal distribution at the phase transition with medium social stress (k = 2). Behavioral phase transition occurs between stable and unstable society induced by collective behavior. c. Bi-modular distribution with high social stress (k = 2.5). The society splits into two opposing groups under low social temperature and strong social interactions in unstable society.

Deterministic models are used by Keynesian economists for endogenous mechanism of business cycles, such as the case of the accelerator-multiplier model. The stochastic models are used by the Frisch model of noise-driven cycles that attributes external shocks as the driving force of business fluctuations. Since 1980s, the discovery of economic chaos and the application of statistical mechanics provide more advanced models for describing business cycles. Graphically, The steady state of probability distribution function in socio-psychological model of collective choice. Here, “a” is the independent parameter; “b” is the interaction parameter. a Centered distribution with b < a (denoted by short dashed curve). It happens when independent decision rooted in individualistic orientation overcomes social pressure through mutual communication. b Horizontal flat distribution with b = a (denoted by long dashed line). Marginal case when individualistic orientation balances the social pressure. c Polarized distribution with b > a (denoted by solid line). It occurs when social pressure through mutual communication is stronger than independent judgment. Numerical 1 autocorrelations from time series generated by random noise and harmonic wave. The solid line is white noise. The broken line is a sine wave with period P = 1.

Linear harmonic cycles with unique frequency are introduced in business cycle theory. The auto-correlations from harmonic cycle and white noise are shown in the above figure. Auto-correlation function from harmonic cycles is a cosine wave. The amplitude of cosine wave is slightly decayed because of limited data points in numerical experiment. Auto-correlations from a random series are an erratic series with rapid decade from one to residual fluctuations in numerical calculation. The auto-regressive (AR) model in discrete time is a combination of white noise term for simulating short-term auto-correlations from empirical data.

The deterministic model of chaos can be classified into white chaos and color chaos. White chaos is generated by nonlinear difference equation in discrete-time, such as one-dimensional logistic map and two-dimensional Henon map. Its autocorrelations and power spectra look like white noise. Its correlation dimension can be less than one. White noise model is simple in mathematical analysis but rarely used in empirical analysis, since it needs intrinsic time unit.

Color chaos is generated by nonlinear differential equations in continuous-time, such as three-dimensional Lorenz model and one-dimensional model with delay-differential model in biology and economics. Its autocorrelations looks like a decayed cosine wave, and its power spectra seem a combination of harmonic cycles and white noise. The correlation dimension is between one and two for 3D differential equations, and varying for delay-differential equation. History shows the remarkable resilience of a market that experienced a series of wars and crises. The related issue is why the economy can recover from severe damage and out of equilibrium? Mathematically speaking, we may exam the regime stability under parameter change. One major weakness of the linear oscillator model is that the regime of periodic cycle is fragile or marginally stable under changing parameter. Only nonlinear oscillator model is capable of generating resilient cycles within a finite area under changing parameters. The typical example of linear models is the Samuelson model of multiplier-accelerator. Linear stochastic models have similar problem like linear deterministic models. For example, the so-called unit root solution occurs only at the borderline of the unit root. If a small parameter change leads to cross the unit circle, the stochastic solution will fall into damped (inside the unit circle) or explosive (outside the unit circle) solution.

# Finsler Space as a Locally Minkowskian Space: Caught Between Curvature and Torsion Tensors. The extension of Riemannian “point”-space {xi} into a “line-space” {xi, dxi} make things clearer but not easier: how do you explain to a physicist a geometry supporting at least 3 curvature tensors and five torsion tensors? Not to speak of its usefulness for physics! Fortunately, the “impenetrable forest” by now has become a real, enjoyable park: through the application of the concepts of fibre bundle and non-linear connection. The different curvatures and torsion tensors result from vertical and horizontal parts of geometric objects in the tangent bundle, or in the Finsler bundle of the underlying manifold.

In essence, Finsler geometry is analogous to Riemannian geometry: there, the tangent space in a point p is euclidean space; here, the tangent space is just a normed space, i.e., Minkowski Space. Put differently: A Finsler metric for a differentiable manifold M is a map that assigns to each point x ∈ M a norm on the tangent space TxM. When referred to the almost exclusive use of methods from Riemannian geometry it means that this norm is demanded to derive from the length of a smooth path γ : [a, b] → M defined by ∫ab ∥ dγ(t)/dt ∥ dt. Then Finsler space becomes an example for the class of length spaces.

Starting from the length of the curve,

dγ(p, q):= ∫pq Lx(t), dx(t)/dt dt

the variational principle δdγ(p, q) = 0 leads to the Euler-Lagrange equation

d/dt(∂L/∂x ̇i) – ∂L/∂xi = 0

which may be rewritten into

d2xi/dt2 + 2Gi(xl, x ̇m) = 0

with Gi(xl, x ̇m) = 1/4gkl(-∂L/∂xl + ∂2L/∂xl∂x ̇m), and 2gik = ∂2L/∂x ̇l∂x ̇m, gilgjl = δij. The theory then is developed from the Lagrangian defined in this way. This involves an important object Nil := ∂Gi/∂yl, the geometrical meaning of which is a non-linear connection.

In general, a Finsler structure L(x, y) with y := dx(t))/dt = x ̇ and homogeneous degree 1 in y is introduced, from which the Finsler metric follows as:

fij = fji = ∂(1/2L2)/∂yi∂yj, fijyiyj = L2, yl∂L/∂yl = L, fijyj = L∂L/∂yi

A further totally symmetric tensor Cijk ensues:

Cijk := ∂(1/2L2)/∂yi∂yj∂yk

which will be interpreted as a torsion tensor. As an example of a Finsler metric is the Randers metric.

L(x.y) = bi(x)yi + √(aij(x)yiyj)

The Finsler metric following is

fik = bibk + aik + 2b(iak)lyˆl − aillakmm(bnn)

with yˆk := yk(alm(x)ylym)−1/2. Setting aij = ηij, yk = x ̇k, and identifying bi with the electromagnetic 4-potential eAi leads back to the Lagrangian for the motion of a charged particle.

In this context, a Finsler space thus is called a locally Minkowskian space if there exists a coordinate system, in which the Finsler structure is a function of yi alone. The use of the “element of support” (xi, dxk ≡ yk) essentially amounts to a step towards working in the tangent bundle TM of the manifold M.