In thermodynamics, surroundings entropy increase is an oft-used postulate, particularly used in life thermodynamics, which states that if living structures represent local pockets of entropy decrease (regions of order) then, in accordance with the second law, there must have been a greater increase in entropy (regions of disorder) in the surroundings outside of the system.

Overview
In 1947, English thermodynamicist Alfred Ubbelohde stated the argument as such: [1]

“Living organisms are characterized thermodynamically not by any vital power of selection of individual molecules, but by the fact that the organism considered as a unit is continually effecting processes, in which the entropy decreases, at the expense of rather greater compensating increases of entropy in the surroundings.”

The proof of the postulate, commonly found, argues that: [2]

(a) the universe can be treated as an "isolated" system, such that:

$dS_{universe} = dS_{isolated} \,$

(b) that the second law for isolated system stipulates that:

$dS_{isolated} \ge 0$

(c) that a decrease in the value of entropy in a system is representative of "order":

$dS \downarrow = order \,$

(d) that an increase in the value of entropy in a system is representative of "disorder":

$dS \uparrow = disorder \,$

(e) that a living entity (living system), such as human being, represents an ordered system:

= Ordered system

(f) that, according to the principle of additivity of entropy, the entropy of system (e), contained within the surroundings of the universe (which is assumed isolated as a whole), relates to the entropy of the universe, such that: [3]

$dS_{universe} = dS_{system} + dS_{surroundings} \,$

	= Ordered system

(g) that with substitution of points (a) and (b), part (f) becomes:

$dS_{system} + dS_{surroundings} \ge 0 \,$

(h) that, in the process of the formation of a living structure, according to the logic of points (c) and (e), there would have been:

$dS_{system} \downarrow \,$

(i) that in order for expression (g), in conjunction with (h), to hold one must conclude that:

$dS_{surroundings} \uparrow \uparrow \,$

It is difficult to find exactly where the proof originated but into the 1970s it gained momentum as a sort of assumed truth.

In 1977, American biologist William DeWitt, for instance, on the topic of cellular energetics, stated that: [4]

“If the entropy of a system decreases, then there must be a larger increase in the entropy of the surroundings, such that the net result of the process is an increase in the entropy of the universe.”

(add discussion)

Difficulties
The central difficulties in the above proof are, firstly, that in point (i) it is inconceivable to think that one can measure changes in entropy (or of heat flow) in the infinit space of the surroundings, which are obviously open to any number of possible changes, to corroborate proof; second, that in point (a) we do not know if the universe is isolated; third, that the suppositions involved in points (c) and (e) are rather unsubstantiated in themselves.

References
1. Ubbelohde, Alfred René. (1947). Time and Thermodynamics, (section “Experimental Aspects of the Relation between Thermodynamics and Life”, pgs. 100-05). Oxford University Press.
2. Singh, Onkar. (2006). Engineering Thermodynamics (pg. 136). New Age International.
3. Aleksishvili, Maka Sidamonidze, Shota. (2002). Problems in Chemical Thermodynamic (pg. 112). World Scientific.
4. DeWitt, William. (1977). Biology of the Cell: An Evolutionary Approach (pg. 88). Saunders.