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I'm wondering if there are any good indicators on how even the heat is distributed on an object (for simplicity, a flat object maybe)? What are the possibly reasonable ways to maximize the evenness if varying the shape of the object is allowed.

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Are you asking about experimental ways to measure whether heat is evenly distributed, or theoretical ways to define whether heat is evenly distributed? – Nathaniel Mar 5 at 3:52
Statistically the longer you wait the more uniform the object will likely by. If you're talking about a real-world object influenced by the environment then I suspect you'd need a supercomputer to model the heat flow over time with any significant accuracy. – Brandon Enright May 4 at 4:08

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Heat is a process quantity, not a system quantity, and there is no indicator that you can attach to the object, but you could use a thermometer to record the distribution of temperatures of the object at different times and then infer the heat involved from some model (e.g. Fourier law).

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