This is really where you are losing me. Even if this gradient you describe is causing organic molecules to 'fall' towards the cell membrane at a rate faster than would be normal without the existence of life, how does that alter either rate of genomic change or external survival pressures such as predation - and in turn, hasten evolution?
In the case of the exterior water and the cell membrane, I was talking more about the evolutionary state called abiogenesis, with the improved flux of organics speeding up whatever is going on inside. To addess those other points of evolution such as genone, we need to looked at the inside membrane boundary conditions. In this case, potassium accumulates. Potassium has the opposite effect as sodium on water or is a kaotrope; increases the disorder/ entropy in the water. The outside water pulls and the inside water pushes, with one proportional to the other. This allows the outside to impact the inside and the inside the outside.
The concept of entropy tends to create a conceptual problem for most people. Although we define entropy in a certain way, entropy is also lumped into other variables, making it harder to see other important aspects of entropy.
A good example of this lumping has to do with weather. A low pressure system results from water vapor in the atmosphere condensing into droplets. This is why low pressure and rain go together. When the water vapor condenses, according to PV=nRT the number of moles of gas decreases in the local air, since we lose the partial pressure of the water vapor. With V,T and R constant the pressure will drop. If you even did canning, when the jars cool. the hot moist air inside the the jar condenses and pulls a vacuum, since we lose partial pressure.
When water condenses into rain droplets, the entropy of the water vapor will decrease, since the water goes from the disorder of the gas phase into the order of the smaller volume of a liquid. This loss of entropy within the air, lowers the pressure in the air. We don't normally call this change a loss of entropy. Who would figure pressure drop in weather and entropy are related. We tend to equate entropy to the diversity in clouds, which is more cosmetic and not pushing a weather system.
There are many such examples of entropy lumped into other variables. To add these to the standard definition, I prefer the expression "degrees of freedom" to better define entropy. This includes all the randomess of the clouds as well as the pressure drop. Condensing water vapor into rain lowers the degree of freedom of the water vapor. The wind coming in from higher pressure system toward lower pressure gains entropy or the air in thw wind will gain degrees of freedom, since now it can move, in a semi-directed way, over the distance between the high and low pressure. It would not have that freedom without the low pressure or entropy drop in the water vapot. But also it loses some degree of freedom since it may move directly toward the low pressure and not be able to meander. Or it might form the order of a tornatoe which restricts its freedom.
If we go back to the outside of the membrane, the induction of the membrane into lower entropy, due to the cationic segregration results in the sodium cations on the outside. The sodium cations, by being a kosmotrope will lowers the entropy in the adjacent water. We now have sort of our pseudo-low pressure/entropy system that is causing higher entropy flow patterns in the local water toward the lower entropy.