Why does the steady state exist In thermal conduction? In thermal conduction,why doesn't transient state go on continuously,why does steady state exist?why after some considerable time A metal bar sttached to a sink and source comes to steady state where the the temperature of different points on bar is different but it doesn't change with time?I want to know what actually happens at molecular level?
 A: Heat flow is governed by the heat equation
$$\frac{\partial u}{\partial t}=D\frac{\partial^2 u}{\partial x^2}$$
where $u(x,t)$ is the temperature at a point and $D$ is a constant. The term $\frac{\partial^2 u}{\partial x^2}$ describes the curvature of $u$ at a point. The reason that the heat equation goes to a steady state is that moving in the direction of curvature makes the curvature smaller. It will act something like this

In the left bump the curvature is negative and in the right bump the curvature is positive. We can show this slightly more rigorously. We know the function $\sin kx$ is an eigenfunction of $\frac{\partial^2 u}{\partial x^2}$ so let's guess a solution of the form
$$u(x,t)=A(t)\sin kx.$$
Plugging this in the heat equation gives
$$A'(t)=-Dk^2A(t)$$
which has solution
$$A(t)=e^{-Dk^2t}.$$
Using a Fourier transform we can write a general function $u$ as a sum of infinitely many sines and cosines. Since we now know how each of these functions behaves we can predict how the function as a whole will behave. The amplitude of each sine wave goes down over time and the sines with the smallest wave length (small wave length means large $k$) disappear the fastest.
I didn't take into account boundary conditions here so if you want to predict what happens when you keep two points at a fixed temperature you would have to do some more work. But the result is the same: curvature decreases over time.
A final note on what happens on a molecular level. The important thing that happens on a molecular level is that heat tends to spread out. This makes the heat diffuse: the heat equation is mathematically the same as the diffusion equation.
A: 
In thermal conduction, why doesn't transient state go on continuously,
why does steady state exist?

As you already know, steady state conditions exist when the temperatures in the bar are no longer changing (increasing) in time. For the transient state to go on continuously, the temperatures in the bar would need to continually increase in time. But the temperature in the bar cannot exceed the temperature of the heat source, $T_H$. To do so would require heat to spontaneously flow from a lower temperature region to a higher temperature region in violation of the second law of thermodynamics.
Bottom line: Temperatures cannot continue to rise because the maximum possible temperature is limited to the temperature of the heat source.
Hope this helps.
