The geometric and exponential distributions are said to be memoryless. How much memory do some other distributions require? (Intended as a serious question.) https://en.wikipedia.org/wiki/Memorylessness …
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Relates to a thing I was wondering about: suppose a process runs for time T before terminating and we know distribution of T. What can we infer about internals of system knowing nothing other than this distribution? If not exponential can we can infer it has some internal state
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Concrete(ish) example: graphics people have started using non-exponential free flight times for photons. Made no sense when I heard it. "Where's the state?" I asked. But in non-homogeneous medium the photon position is state participating in the process https://benedikt-bitterli.me/nec/
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Another example: a non-exponential failure time for a machine tells you its state is changing, eg. from wear and tear. Exponential suggests (to me at least) wear-and-tear isn't the issue.
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Replying to @sigfpe
Fun granular physics facts: if you let sand flow out of a bucket with a hole in the bottom, the distribution of time until the hole clogs is exponential. If you start with a bucket with a clogged hole and vibrate it, the time until unclogging is heavy-tailed ... 1/2
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Replying to @blockspins @sigfpe
in other words, the longer a clog has survived, the more likely it is to continue living for a long time. See this paper https://arxiv.org/abs/1711.04455 which gives a model for all this in terms of exploring an energy landscape (a bit strange because the system is nonequilibrium). 2/3
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I first heard about this stuff from @CharlesCThomas_ when I was in grad school: he did a bunch of neat experiments on *tilted* granular hoppers! https://arxiv.org/abs/1410.0933 https://arxiv.org/abs/1206.7052
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Replying to @blockspins @CharlesCThomas_
I find granular physics scary! I like to think that you can do physics on a continuum by deriving the physics for an infinitesimal element and then integrating. But this seems to fail for granules. Leaves room for good experiments though
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