I've certainly never seen one. There is a little leakage through from the collector to the emitter, but with modern silicon BJTs you can usually ignore it. When the base is forward biased by taking it about 600 mV above the emitter, charge carriers are launched into the base region. Since the base region is narrow, only a small proportion of these carriers are snared and become the base current. The vast majority shoot through into the collector, to form the collector current Ic.
T is the absolute temperature in Kelvin.
Every transistor obeys it with startling accuracy over nine or ten decades of Ic, which is a pretty broad hint that we are looking at the fundamental mechanism. In contrast, beta varies with Ic, temperature, and just about everything else. This is called Early Effect, after Dr Early, and has nothing to do with timing or punctuality. Never design circuits that depend on beta. Here are some of the factors that affect beta.