Why is an isothermal process not unique in nature? To be clear, I have got two doubts,


One, about the non-uniqueness of an isothermal process &amp;
Two, about the uniqueness of an adiabatic process.


The first doubt

The Wikipedia page of isothermal process states that:


  For an adiabatic process, in which no heat flows into or out of the gas because its container is well insulated, Q = 0. If there is also no work done, i.e. a free expansion, there is no change in internal energy. For an ideal gas, this means that the process is also isothermal.[4] Thus, specifying that a process is isothermal is not sufficient to specify a unique process.


Now I am confused about the meaning of the last line.


  Thus, specifying that a process is isothermal is not sufficient to specify a unique process.


I am not able to understand the significance of that last statement. Does it mean that any two states, say A &amp; B, can be connected by more than one isothermal processes?

The second doubt

Similarly, I had read in my textbook that an adiabatic process is unique in nature. That is, between any two states only one adiabatic process can exist. To say the same thing in other words, 
if two thermodynamic states are connected by a reversible adiabatic process, then the same two states cant be connected by an irreversible adiabatic process

Kindly help me understand these statements with the help of some PV/VT/PT diagrams.

Any help would be deeply appreciated.
(1) An isothermal process is one during which the system temperature $T=constant$. For an ideal satisfying $pV=RT$ this means that $pV=constant$ irrespective of how you achieve it. You can go up and down, back and forth on the same isothermal, but it is still an isothermal process having the same temperature. So it is not a unique process between two fixed points.

(2) The uniqueness of the adiabatic process is true only for a simple system, that is one whose state can be described by temperature and one additional "mechanical" parameter, e.g., for a gas $p = p(T,V)$, or a magnet $M=M(T,B)$, etc. It is definitely not true for more complicated systems, for example a gas with magnetic properties $p=p(T,V,B)$. Instead what is true is that through every equilibrium point there is a unique adiabatic surface (not a curve) determined by a pair of variables say $T,V$ or $V,B$ or $T,B$ such that any curve lying in that surface is an adiabatic process. That such adiabatic surfaces exist is the basis of one of the mathematical proofs of the existence of the entropy function and this quite famous proof is due to Caratheodory.