= Heat bath
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= Thermal reservoir
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A heat bath, or thermal reservoir, is an idealized system so large that exchanging a finite amount of <heat> does not appreciably change its <temperature>.
The ideal <heat bath> supplies a fixed absolute <temperature> $T_b$ while its <internal energy> changes through <heat transfer>. A real body's suitability as a <heat bath> depends on its <heat capacity> relative to the exchange being studied, and on how quickly its internal <temperature gradients> disappear. Large physical size alone supplies neither condition. A <finite heat bath> makes the approximation quantitative; a <regulated calibration bath> instead uses <feedback control> to maintain its <temperature> by exchanging <energy> with external equipment.
For a weakly coupled system that exchanges <energy> but no matter with an ideal <heat bath>, the <canonical ensemble> assigns the <Boltzmann distribution>
$$
p_n=\frac{e^{-E_n/(k_BT_b)}}{Z},\qquad Z=\sum_m e^{-E_m/(k_BT_b)}.
$$
Here $E_n$ is a <microstate>'s <energy>, $k_B$ the <Boltzmann constant> and $Z$ the <partition function>. The fixed <temperature> removes the need to track every exchange with the environment. A substantially changing bath <temperature> no longer gives this one fixed <Boltzmann distribution>.<Statistical Physics by David Tong/1 The Fundamentals of Statistical Mechanics>[Statistical Physics, §1.3, printed pp. 17–18, equation (1.19): reservoir assumption and canonical probabilities]{ref}
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