Non-linear Differential Equations and Psuedo-randomness

In summary, the conversation discusses the use of statistical approaches to deal with turbulence and the question of whether the solutions to the Navier-Stokes equation and other chaotic systems are pseudorandom. It is mentioned that some solutions in 3D may have a global attractor and be pseudorandom, but there is no general theorem for this. The Navier-Stokes equation is also mentioned as a million dollar problem and the possibility of connections between ergodicity and pseudorandomness for other systems is raised.
  • #1
linford86
14
0
I was thinking about the non-linear Navier-Stokes equation this morning and was briefly browsing a text on the subject. I'm aware that one popular approach to dealing with turbulence is to take averages and look at correlators (which, in turn, can be related to field theory.) Now, one thing which strikes me as odd about this statistical approach to dealing with turbulence is that the Navier-Stokes equation is fully deterministic; given some set of initial and boundary conditions, the entire time evolution of the system is determined.

So, I have a question about the Navier-Stokes equation: is it known that the solutions are psuedo-random for high Reynold's number? Of course, one can extend this more generally to dynamical systems with positive Lyapunov exponents -- are the solutions to such systems (e.g., chaotic differential equations) known to be psuedo-random? If so, it would appear to be natural to attack them statistically since they would pass tests for randomness.
 
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  • #2
Many solutions having a global attractor in 3D are pseudo-random. Howver, I haven't heard of a general theorem.
 
  • #3
Interesting. Do you have any examples? I'd like to know more about this subject.
 
  • #4
Check the Lorenz attractor ,for instance.
There are a few theorems connecting ergodicity & randomness, but the general case of the Navier-Stokes equation is monstrous. It's not known if a general
solutions exists ( a million dollar problem) , let alone the contingent property of pseudorandomness.
 
  • #5
Yes, I'm aware of the related Millenium Problem. At any rate, does anyone have any information connecting ergodicity and psuedo-randomness for other systems? That seems intriguing to me.
 

What are non-linear differential equations?

Non-linear differential equations are mathematical equations that involve variables raised to a power, or multiplied or divided by each other. They are called 'non-linear' because the relationship between the variables is not a straight line.

How are non-linear differential equations different from linear differential equations?

Linear differential equations have a relationship between the variables that can be represented by a straight line, while non-linear differential equations have a more complex relationship that cannot be represented by a straight line.

What is the significance of psuedo-randomness in differential equations?

Pseudo-randomness is important in differential equations because it allows for the simulation of real-world systems with unpredictable or chaotic behavior. It also helps to generate more accurate and realistic results in mathematical models.

What methods are used to solve non-linear differential equations?

There are various methods for solving non-linear differential equations, such as numerical methods like Euler's method or Runge-Kutta methods, as well as analytical methods like separation of variables or substitution.

How are non-linear differential equations used in scientific research?

Non-linear differential equations are used in a wide range of scientific fields, such as physics, biology, chemistry, and engineering, to model and predict the behavior of complex systems. They are also used to study chaotic systems and understand the dynamics of natural phenomena.

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