What is the relationship between state variables and conservative force fields? State variables are defined as being independent of path, just like conservative force fields. Is there any relationship between state variables and conservative force fields? Is it because there is no direct relation to $\Delta x$ for state variables like temperature and pressure?
 A: 
Is there any relationship between state variables and conservative
force fields?

State variables are system properties. Forces are not system properties. The relationship between forces and state variables is that forces can transfer energy in the form of work (the other being heat) and the energy transfer can result in changes in state variables .

Is it because there is no direct relation to $\Delta x$ for state
variables like temperature and pressure?

Work, $F\Delta x$, transfers energy to or from a system and the energy transfer can cause changes in state variables of that system (temperature, pressure, volume, internal energy, etc.)
The difference between work done by conservative forces vs work done by non conservative forces is the former is path independent while the latter can be path dependent. So in cases where energy transfer is strictly in the form of work (no heat transfer), the transfer of energy due to conservative forces exactly equals the change in internal energy, a system property.
Hope this helps.
