I have been reading the book "The Black Hole War," by L.Susskind, which states:
Temperature is the increase of a system's energy when an entropy bit is added.
Until now I was thinking that, on the contrary, it is the entropy that is increased when energy is added. No matter how you define entropy as hidden information or diffusion of energy or log of microstates or as heat divided by absolute temperature, the arrow of causality is from energy to entropy and not the other way around. I also don't understand why adding of entropy by itself increases energy.
What is the meaning of this definition?
Answer
The sentence is really just translating the equation
$$ dU = TdS $$
into words, although it is also neglecting the other components of energy changes, $-PdV + \mu dN$, which are the energy changes due to changes in volume and chemical composition.
If you rearrange the equation and hold volume, pressure and composition constant, you get:
$$ \frac{dU}{dS}\biggr\rvert_{V,P,N} = T $$
And that is what the sentence is saying. Temperature is the change in energy due to the change in entropy. And since there is no negative sign, it is phrased as a positive -- energy increases when entropy is added.
Now, if temperature is constant, then what you said in your question is correct -- if you double the energy, the entropy will also double. But the reverse is true as well. For a fixed temperature, if you double the entropy, the energy doubles also.
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