an exponential decrease in measurement error will only buy you a linear increase in how long that simulation is good for.
True, and in the real world attempts to measure with extreme precision eventually hit limits imposed by quantum mechanics. Quantum systems are unpredictable in a way that has nothing to do with chaos theory, but that cashes out to injecting tiny amounts of randomness in basically every physical system. In a chaotic system those tiny perturbations would eventually have macroscopic effects, even in the absence of any other sources of error.
I don’t know the exact values Lorenz used in his weather simulation, but Wikipedia says “so a value like 0.506127 printed as 0.506”. If this were atmospheric pressure, we’re talking about a millionth decimal place precision. I don’t know what exerts 0.000001 Pa of pressure or to what such a teeny pressure matters.
an exponential decrease in measurement error will only buy you a linear increase in how long that simulation is good for.
True, and in the real world attempts to measure with extreme precision eventually hit limits imposed by quantum mechanics. Quantum systems are unpredictable in a way that has nothing to do with chaos theory, but that cashes out to injecting tiny amounts of randomness in basically every physical system. In a chaotic system those tiny perturbations would eventually have macroscopic effects, even in the absence of any other sources of error.
I don’t know the exact values Lorenz used in his weather simulation, but Wikipedia says “so a value like 0.506127 printed as 0.506”. If this were atmospheric pressure, we’re talking about a millionth decimal place precision. I don’t know what exerts 0.000001 Pa of pressure or to what such a teeny pressure matters.
That’s the point. Nobody thought such tiny variations would matter. The fact that they can matter, a lot, was the discovery that led to chaos theory.