Statistics question on Poisson distribution?

I understand how Poisson distribution can be used to model infrequent events. Let me give a concrete example of the type of calculation I would like to perform.

Suppose we know the rate at which particles arrive at a specified location and suppose the rate is small. We define a collision to be the arrival of two or more particles at the location within the same minute. We can use the Poisson distribution to calculate the probability of a collision during a specified minute. How can you determine the probability of at least one collision during a particular 24 hour period. I do not want to break the day up into one minute intervals. I want to use a continuous time distribution.