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$$P(\phi,\Pi) = e^{-\beta ({1\over 2} |\Pi|^2 + {1\over 2} |\nabla \phi|^2 + V(\phi)} $$

$$ P(\phi,Pi) = e^{ - \beta (\sum_k {1\over 2} |\Pi(k)|^2 - {1\over 2} k^2|\phi(k)|^2 - V(\phi)} $$

$$P(\phi,\Pi) = e^{-\beta (\sum_i {1\over 2} |\Pi_i|^2 + {1\over 2} |\nabla \phi_i|^2 + V(\phi) )} $$

$$ P(\phi,Pi) = e^{ - \beta (\sum_{ik} {1\over 2} |\Pi_i(k)|^2 - {1\over 2} k^2|\phi_i(k)|^2 - V(\phi)} $$

The delta function enforces translation invariance, and it is important, because it says that modes of size |k| can only interact with each other to make modes of size bounded by 4|k|, at worst, and typically only of size 2k. This means that the flow of energy is sort-of local in k-space, because the mixing nonlinearity can't push the energy in one step from small |k| to very large |k|, it can only add a factor of $log(2)$ to the log of the size of k. This is obviously true for any polynomial term in a nonlinear equation, and it is more true for more relevant terms.

The delta function enforces translation invariance, and it is important, because it says that modes of size |k| can only interact with each other to make modes of size bounded by 3|k|, at worst, and typically only of size 2k. This means that the flow of energy is sort-of local in k-space, because the mixing nonlinearity can't push the energy in one step from small |k| to very large |k|, it can only add a factor of $log(2)$ to the log of the size of k. This is obviously true for any polynomial term in a nonlinear equation.

The exponential suppression of large energies is easy to understand statistically. There is a conserved quantity, the energy, which is global. In order to absorb a unit of energy, you have to pay a probability cost which is uniform, but otherwise, there is no constraint on the motion. This is the maximum-entropy interpretation of the Boltzmann state--- each subsystem pays a probability cost for absorbing a unit of energy, and this probability cost is adjusted by making sure the total energy is whatever it is. The maximum entropy distribution for any conserved quantity is found by imposing a probability cost for absorbing each unit of the conserved quantity, so that each subsystem is only probabilistically constrained by the amount of each conserved quantity that it has. The log of the probability cost for the quantity is the thermodynamically conjugage variable (or, rather, it would be, if thermodynamics developed logically rather than historically. In actual life, people multiply all the thermodynamic conjugate quantities by the temperature for no good reason, to turn the more fundamental extra log-probability quantity into a less fundamental quantity which is the extra free energy per unit conserved quantity, so that the thermodynamically conjugate quantity to U has units of energy per unit U, rather than (quantumly dimensionless) entropy per unit U. I try not to use this otherwise universal convention, because I think it is wrongheaded. Also, it is good use $\beta$ instead of T most of the time, since $\beta$ is the thermodynamically conjugate variable to E.)

This process is well known in mathematics--- it is the 3n+1 procedure, the Collatz problem. It is a simple consequence of eventual randomization that the 3n+1 Collatz conjecture is true, that all finite patterns reach "1" eventually (because all infinite random bit strings are shifted to the left after a many iterations with probability 1). But to prove this conjecture rigorously is well beyond current mathematical methods. So proving that a deterministic system turns random in any meaningful way is generally extremely difficult. Even so, seeing that it turns random is generally not difficult-- you can identify the stochasticity by eye and by simple statistical tests. Further, it is often not difficult to identify what the correct probability distribution should be, once it turns random, just by identifying the conserved quantities in the problem, and making a distribution function from these conserved quantities which is preserved under time evolution. This is the source of many conjectures.

The exponential suppression of large energies is easy to understand statistically. There is a conserved quantity, the energy, which is global. In order to absorb a unit of energy, you have to pay a probability cost which is uniform, but otherwise, there is no constraint on the motion. This is the maximum-entropy interpretation of the Boltzmann state--- each subsystem pays a probability cost for absorbing a unit of energy, and this probability cost is adjusted by making sure the total energy is whatever it is. The maximum entropy distribution for any conserved quantity is found by imposing a probability cost for absorbing each unit of the conserved quantity, so that each subsystem is only probabilistically constrained by the amount of each conserved quantity that it has. The log of the probability cost for the quantity is the thermodynamically conjugage variable (or, rather, it would be, if thermodynamics developed logically rather than historically. In actual life, people multiply all the thermodynamic conjugate quantities by the temperature for no good reason, to turn the more fundamental extra log-probability quantity into a less fundamental quantity which is the extra free energy per unit conserved quantity, so that the thermodynamically conjugate quantity to U has units of energy per unit U, rather than (dimensionless, quantumly additively unambiguous) entropy per unit U. I try not to use this otherwise universal convention, because I think it is wrongheaded. Also, it is good use $\beta$ instead of T most of the time, since $\beta$ is the thermodynamically conjugate variable to E.)

This process is well known in mathematics--- it is the 3n+1 procedure, the Collatz problem. It is a simple consequence of eventual randomization that the 3n+1 Collatz conjecture is true, that all finite patterns reach "1" eventually (because all infinite random bit strings are shifted to the right after a many iterations with probability 1). But to prove this conjecture rigorously is well beyond current mathematical methods. So proving that a deterministic system turns random in any meaningful way is generally extremely difficult. Even so, seeing that it turns random is generally not difficult-- you can identify the stochasticity by eye and by simple statistical tests. Further, it is often not difficult to identify what the correct probability distribution should be, once it turns random, just by identifying the conserved quantities in the problem, and making a distribution function from these conserved quantities which is preserved under time evolution. This is the source of many conjectures.