# Random physics

Alberto Verga, research notebook


## Dynamical systems

The state of a physical system $$O$$ is determined by a set of variables $$X$$ and parameters $$M$$, together noted $$o=\{X,M\}$$; $$o$$ represents one state of the system $$O$$. Physical laws do not depend on units, therefore the set $$M$$ is nondimensional.

In a dynamical system we distinguish the time parameter $$t$$ on which the state depends $$o = o(t) \in O$$. If $$t \in T = \mathbb{R}$$ the system’s evolution is continuous; if $$t \in T = \mathbb{Z}$$ it is discrete. The time trajectory $$o(t)$$ is generated by the one-parameter flow operator $$F_t$$:

$$o(t) = F_t \, o(0) \,, \quad F_{t_1} \circ F_{t_2} = F_{t_1 + t_2}$$

where $$o(0)$$ is the initial state (the state at $$t=0$$). At variance with variables $$X$$ that depend on time, the set $$M$$ of parameters on which $$F$$ depends, are considered fixed (independent of $$t$$)

$$o(t) \equiv X_M(t)\,.$$

The history $$X= X_M(t)$$ is called the “trajectory” or orbit of the dynamical system and the space spaned by $$X$$ the “phase space”. If $$F$$ is invertible, then the flow has a group structure.

### Examples:

1. Newton dynamics is governed by the law relating force with acceleration,

$$m\frac{\D^2 \boldsymbol{x}}{\D t^2} = - \Dd{\boldsymbol{x}} V(\boldsymbol{x}) \,,$$
which gives the trajectory $$\boldsymbol{x} = \boldsymbol{x}(t)$$ for a particle of mass $$m$$ in a potential $$V$$. This ordinary differential equation trivially generalizes to a system of particles. In terms of the phase space vector $$z=(x,p)$$ and the Hamiltonian $$H=H(z)$$ the above equation can be written as:
$$\dot{z} = \Omega \frac{\partial }{\partial z} H$$
where $$\Omega$$ is the symplectic matrix
$$\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\,,$$
here $$1$$ is the identity matrix, based on the dimension of the configuration space $$\mathrm{dim}\,x$$, and $$\partial/\partial z$$ is the gradient vector.

2. Quantum mechanics. A quantum system defined by a Hamiltonian $$H$$ evolves in time according to the Schrödinger equation,

$$|\psi(t) \rangle = U(t,0) | \psi(0) \rangle \,,$$
where $$| \psi \rangle$$ is the quantum state (a vector in Hilbert space) and the operator $$U$$ is given by ($$\hbar = 1$$),
$$U = \E^{-\I H t},$$
in the case of $$H$$ independent of time. As a consquence of the hermicity of $$H=H^\dagger$$, the evolution is unitary $$U^{-1}=U^\dagger$$ (even if $$H=H(t)$$). Quantum dynamics is invertible.

## The logistic map

A simple and rich dynamical system is the logistic map,

$$x_{n+1} = \mu x_n (1 - x_n)\,, \quad n = 0,1,2,\ldots$$

Depending on the value of the parameter $$\mu \in [0,4]$$, the iterates $$x_n$$ can tend towards a fixed point $$x^* = x_{n+1} = F_1(x_n) = x_n$$, a periodic orbit $$x_{n+p} = F_p(x_n) = x_n$$, of period $$p$$, or for most values $$\mu > \mu_\infty = 3.57$$, chaotic. The stability of a periodic orbit is determined by the value of the derivative of the map at the fixed point (Jacobian $$\D x_{n+1}/ \D x_n$$); for $$D[F_p](x^*)<1$$ the orbit is stable.

For instance, the $$p=1$$ fixed point becomes unstable at $$\mu = 3$$, from which a double periodic orbit $$p=2$$ sets in. Transition toward chaos is through period-doubling bifurcations, whose accumulation point ($$p=2^n$$ with $$n \rightarrow \infty$$) is just $$\mu_\infty$$.

Show that the dyadic map, $$x_{n+1} = 2 x_n \mod{1}$$ (Bernoulli shift), is equivalent to the logistic map for $$\mu=4$$ and demonstrate that, in this case, the orbit is given by

$$x_{n+1} = \sin^2 \left( \pi a 2^n \right) \,, \quad x_0 = \sin^2 \pi a$$
if the initial condition is $$x_0$$. We note that for $$\mu=4$$ the logistic map $$F$$ is equivalent to the Bernoulli shift; the Bernoulli shift can be lifted to the complex unit circle: $$B(z) = z^2$$ with $$|z|=1$$. Direct computation of $$F_n(x_0)$$ gives the result.

#### Exercise:

Investigate numerically the logistic map; show empirically the divergence of two initially close trajectories; build the bifurcation diagram; study the histogram of the iterates for $$\mu = 4$$ and compare with a random process.

A simple code:

"""
Logistic map:
logistic_points(mu, n):
x_n values of the logistic map mu*x*(1-x)
"""
def f_logistic(mu, x):
return mu*x*(1-x)

def logistic_points(mu, n = 100):
x = 0.2
#
# filter transient
for i in range(100):
x = f_logistic(mu, x)
#
# iterates
xn = zeros(n)
for i in range(n):
x = f_logistic(mu, x)
xn[i] = x
return xn


## The standard map

The discrete time map defined by,

$$\label{e:sm} x_{n+1} = x_n + p_{n+1}\,, \quad p_{n+1} = p_n + K \sin x_n$$

where $$(x,p)$$ are position (angle) and momentum (action) modulo $$2\pi$$ variables, is a Hamiltonian dynamical system,

$$\label{e:smh} H = \frac{p^2}{2} - K \sum_{n=-\infty}^{\infty} \cos(x - 2\pi nt)$$

as can be easily verified using the Poisson summation formula:

$$\sum_{n=-\infty}^{\infty} \delta(t-n) = \sum_{n=-\infty}^{\infty} \E^{\I 2\pi nt}\,.$$

Equation ($$\ref{e:sm}$$) define the Chirikov standard map, also important in quantum mechanics, which describes the transition to chaos in Hamiltonian systems. The origin of chaos is in the overlap of resonances: when the distance between the separatrices $$\Delta p = 2 \sqrt{K}$$ (the separatrix corresponds to $$E=K$$ line) becomes of the order of the resonance separation $$\Delta \omega = 2\pi$$. This criterion gives $$K_c = \pi^2/4$$; numerical evaluation gives a stochastic threshold at $$K_c \approx 0.98$$.

### Poincaré section and monodromy matrix

The standard map is the Poincaré section of Hamiltonian ($$\ref{e:smh}$$): a section of the phase space transversal to the trajectories, such that a periodic orbit appears as a point (a double periodic orbit appears as two points, etc.).

Consider a general twodimensional $$z=(x,p)$$ Poincaré map $$z \rightarrow z'=P(z)$$; if $$z$$ is a fixed point

$$P(z) = z$$

then

$$\delta z' = P(z+\delta z) - P(z) \,, \quad P(z+\delta z) = z + \D P(z) \delta z\,.$$

where the matrix $$M = \D P (z)$$

$$M = \begin{pmatrix} \frac{\partial x'}{\partial x} & \frac{\partial x'}{\partial p}\\ \frac{\partial p'}{\partial x} & \frac{\partial p'}{\partial p} \end{pmatrix}$$

corresponding to the tangent map, is called the monodromy matrix; it contains information about the stability of the fixed point:

$$\delta z' = M(z) \delta z$$

which is the linearized version of the Poincaré map in the neighborhood of the fixed point $$z$$. Since $$P$$ is an area preserving map (Liouville theorem), the determinant of $$M$$ is one, $$\det \, m = 1$$. In the case of the standard map the explicit form of the monodromy matrix is,

$$M = \begin{pmatrix} 1 & K \cos x \\ 1 & 1 + K \cos x \end{pmatrix}$$

We verify that $$\det M = 1$$. The general form of the eigenvalues is,

$$\lambda_\pm = \frac{1}{2}\left( \mathrm{Tr}\,M \pm \sqrt{|\mathrm{Tr}\,M|^2 - 4} \right) \,,$$

and the corresponding eigenvectors $$v_\pm$$ define two orthogonal directions in the transformed phase space $$(v_-,v_+)$$. Therefore, according to the value of the trace $$\mathrm{Tr}\,M$$, the eigenvalues would be unit complex numbers $$\lambda_\pm = \E^{\pm \I a}$$, with $$0<a<\pi$$ ($$|\mathrm{Tr}\,M|<2$$), real reciprocal numbers $$\lambda_+ = 1/\lambda_-$$ ($$|\mathrm{Tr}\,M|>2$$), or equal $$\lambda_-=\lambda_+=\pm 1$$ ($$|\mathrm{Tr}\,M|=2$$).

For instance, for the fixed point $$x=\pi$$ of the standard map,

$$\lambda_\pm = \E^{\I a}, \quad \cos a = 1 - K/2\,, \; \sin a = \sqrt{K - K^2/4}\,,$$

and, nearby points on a circle around, are mapped to points in the same circle, hence $$x=\pi$$ is an elliptic point. In the case of the fixed point $$x=0$$, stable and unstable manifolds cross, and neighboring points are mapped along hyperbolas, that is, $$x=0$$ corresponds to a hyperbolic point. In fact, the elliptic point becomes a hyperbolic point at $$K=4$$. We can speculate that above this threshold the whole phase space becomes chaotic.

#### Exercice

Investigate the behavior of the standard map as a function of $$K$$

def standard_n(K, N):
N0 = 20
p = linspace(-pi, pi, N0)
x = pi*ones(N0)
#
for n in range(N):
p = mod(p + K*sin(x), 2*pi)
x = mod(x + p, 2*pi)
plot(x, p, 'k.', ms = 0.5, alpha=0.5)


The behavior near $$K=4$$ is interesting:

We observe the disappearance of the elliptic point and the formation of two separated islands within a stochastic sea.