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Steady state with respect to the rod being heated from one end is the state when the temperature of each section of the rod becomes constant.

$\dfrac{\partial p}{\partial t}= 0 $ //where $p$ indicates properties of the substance, here temperature.

Now, it's very similar to equilibrium but I wonder if it's possible to reach this state before equilibrium. If yes, how? Moreover, heat is always going to flow through the cross sections of the rod (before equilibrium) and we cannot prevent it, can we? So is steady state just a hypothetical concept?

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"Equilibrium" refers to the state where there are no gradients in space, and "steady state" refers to the state where there no gradients in time. Steady state is not an hypothetical concept in the sense that achieving steady state does not require that you violate any laws of physics.

You can reach steady state without every reaching equilibrium. If you consider a 1-D heat conduction problem through a slab of homogeneous solid, then given the temperature of its two faces a linear temperature profile is established within the solid after sufficiently long time. This is steady state but not equilibrium. However usually an equilibrium also implies steady state although in theory it need not be so. When an insulated box of gas is allowed to come to equilibrium it attains uniform temperature everywhere, and then this temperature does not change with time either which implies steady state.

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  • $\begingroup$ Please give an example of steady state $\endgroup$
    – user116688
    Commented Aug 17, 2017 at 5:23
  • $\begingroup$ @Abcd I already have. See the statement about 1-D heat conduction. $\endgroup$
    – Deep
    Commented Aug 17, 2017 at 5:24
  • $\begingroup$ How do we establish the "linear temperature profile" ? Also, do you mind including a diagram? $\endgroup$
    – user116688
    Commented Aug 17, 2017 at 5:25
  • $\begingroup$ @Abcd This is staple example of undergraduate thermodynamics textbooks. Refer for example "Thermodynamics" by P.K. Nag. $\endgroup$
    – Deep
    Commented Aug 17, 2017 at 5:27
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    $\begingroup$ @pentane If we take an object with very high conductivity (ideally in the limit $\to\infty$) immersed in a heat bath whose temperature varies slowly. Then at any given instant, temperature of the object would be uniform and equal to temperature of the bath at that instant. It would be in thermal equilibrium at every instant but not in steady state. $\endgroup$
    – Deep
    Commented Aug 18, 2017 at 9:11

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