Hcfyk

Am I correct in saying that the fusion process of the sun is constant, i.e. X amount of fusion happens per day, more or less?
Why does this not speed up, i.e. one fusion event creates energy for two fusion events, etc.?
Does every collision of atom cause a fusion event, or is the probability small for a fusion event to happen thus it's not a runaway reaction?
I have heard that the probability for fusion event to happen is only 1 in 10¹² for every collision.

Am I correct in saying that the fusion process of the sun is constant, i.e. X amount of fusion happens per day, more or less?

Yes, at least over human timescales. You could reasonably expect the fusion rate within the sun to be the same today as a few thousand years ago or into the future, give or take some small fraction.

Why does this not speed up, i.e. one fusion event creates energy for two fusion events, etc.?

The energy released by fusion is quickly distributed as thermal energy in the centre of the sun, and the temperature difference between surface (around 6000K) and centre (estimated 15 million K) drives an energy flow from hot to cold.

Does every collision of atom cause a fusion event, or is the probability small for a fusion event to happen thus it's not a runaway reaction?

Fusion in the sun is not a runaway nuclear reaction (like a critical mass of uranium in a fission reaction).

It is possible in theory to have runaway fusion events, but the pressure and temperature for these to happen are not approached in the core of the sun. For stable stars like the sun, the forces and energy flows are in equilibrium - if the core grew slightly hotter, then the pressure would increase and the star expand slightly against the force of gravity to compensate. Interesting things happen when stars fall out of equilibrium and runaway fusion ignition can happen in some scenarios.

In addition, this equilibrium point moves during the lifetime of a star as its mix of elements changes due to fusion. This is predictable for many stars and forms the basis of the main sequence stars in the Hertzsprung-Russell diagram

I have heard that the probability for fusion event to happen is only 1 in 10^12 for every collision

I don't know the accuracy of that, but it seems reasonable. The definition of "collision" becomes somewhat arbitrary in such a hot dense environment. If you only include approaches close enough to make the strong nuclear force dominate the interaction, the ratio could be higher.

Another fact that I found interesting in the same area is that the power density from fusion - i.e. the Watts per cubic metre of substance - in the sun is roughly the same as that found in a typical compost heap. It is a very different environment to the inside of a fusion reactor experiment or a fusion bomb, which have much higher power densities.

No, the fusion rate of the Sun is not absolutely constant in time. The Sun is gradually becoming more luminous and that luminosity is provided for almost exclusively by fusion in the core. However, the rate of increase is not large, of order 10% per billion years.

The fusion process is extremely slow (and inefficient in terms of energy release per unit volume) - the Sun releases only 250 W/m$^3$ in it's core. The reason for this is that fusion events are extremely unlikely, requiring two protons to overcome the Coulomb barrier between them and for one of the protons to inverse beta-decay into a neutron so forming a deuterium nucleus.

The average lifetime of a proton against this process in the core is $10^10$ years (the lifetime of the Sun), meaning the fusion rate per proton is about $3 times 10^-18$ s$^-1$. We can compare this to a collision rate between protons by assuming an average thermal speed of $v simeq (3k_B T/m_p)^1/2 = 600$ km/s for a core temperature of $15times 10^6$ k, a proton number density of $n_p sim 6 times 10^31$ m$^-3$ in the core and a collisional cross-section of $sigma sim pi (hbar/mv)^2$, where the term in brackets is the reduced de Broglie wavelength. Putting these things together, the collision rate is $n_p sigma v sim 10^12$ s$^-1$.

Thus comparing the two rates, we can conclude that only about 1 in $3times 10^29$ collisions ends up with fusion.

If the fusion rate of the Sun did increase rapidly then what would happen is that the Sun would expand, the core would become less dense and the fusion rate would fall. This basically acts as a thermostat, keeping the Sun at exactly the right temperature to support its own weight and supply the luminosity emerging from its surface.

What's more, the usual explanation for why the fusion doesn't run away is incomplete. The simple story that can't be the full story is that if the fusion happens too fast, heat builds up and creates an overpressure. That overpressure causes expansion, and expansion does work which lowers the temperature and dials back down the fusion until it matches the radiative escape rate.

The reason this is incomplete is that expansion work doesn't induce stability if it occurs only against a fixed external pressure, that amount of work is always insufficient to stabilize it (which leads to "shell flashes" later in the life of a star). The only thing that is capable of stabilizing the fusion is the additional work against gravity, as you can easily see from how gravity gets included in any such analysis. So it must be important that a local runaway has the net result of lifting gas away from the solar center, thereby doing gravitational work-- an important detail normally left out of the explanations. Indeed, it would be more fair to say that solar fusion is stabilized by a combination of expansion work and gravitational lifting.