Dear Rayadurgam,

Achievable Rate, Sum Rate, Ergodic capacity are different parameters used to define the capacity and data rate of the system under consideration. Achievable rate is the maximum number of bits transmitted on the given channel per second. Sum rate is usually used in the multi-channel communication system and is defined as the total capacity from source to destination. However, Ergodic capacity is used for slowly varying channels where the stochastic expectation is explored to find out capacity. All these parameters are related to the mutual information and hence with probability. You can go through the following links for more knowledge

1) https://www.youtube.com/watch?v=Qu-CloVxU2A&t=1s

and

2)

Article Machine learning-based physical layer security: techniques, ...

The ergodic capacity is the theoretically maximum data rate (bit/s) that one communicate at over a fading channel, where the fading processing is an ergodic process. More precisely, this is the largest rate for which the probability of decoding errors goes to zero as the length of the transmission goes to infinity.

An achievable rate is any data rate below the capacity. It is normally associated with a particular (suboptimal but practical) way of transmitting.

The sum rate is the summation of the achievable rates of multiple concurrent transmissions, for example, different users that are multiplexed.