It took several months of googling ;) but I believe you are

Albert Einstein

Someone broke my heart,

OP mentioned in a comment that this means that the answer has been broken down into parts.

Part of me resides in the intersection of Rand al'Thor and the 26

Rand al'Thor himself recognized that the intersection of his name and the name of the alphabet is AL

Sometimes I just need every way of knowing who the fourth to win a race was.

The fourth element on the Periodic Table is Beryllium whose symbol is Be.

Then again sometimes I just run marathons and win silvers and bronzes.

This could clue RT for Rotten Tomatoes, a website that reviews movies that, as the name implies, do not win gold medals. This has nothing to do with marathons (other than maybe a movie watching marathon) but OP said in the comments that a lot of the words are "gibberish" to "maintain the flow". The next line seems to confirm that this may be about movie reviews.

But the sums of my life is the sums of all the races I ran and all the movies I saw.

AL + BE + RT = ALBERT

And that gives me the energy to go ahead and fulfil my dreams.

Albert Einstein discovered the famous equation $E = mc^{2}$ which relates energy to mass and the speed of light.

Who am I? I've been asking the same to myself for over a hundred years lmao

Albert Einstein was born in 1879 and has had over a century to question his identity. If this answer is correct then I hope it helps mend his heart and reaffirm his identity.

The hints:

Albert Einstein had White hair and did physics.

It took several months of googling ;) but I believe you are

Albert Einstein

Someone broke my heart,

OP mentioned in a comment that this means that the answer has been broken down into parts.

Part of me resides in the intersection of Rand al'Thor and the 26

Rand al'Thor himself recognized that the intersection of his name and the name of the alphabet is AL

Sometimes I just need every way of knowing who the fourth to win a race was.

The fourth element on the Periodic Table is Beryllium whose symbol is Be.

Then again sometimes I just run marathons and win silvers and bronzes.

This could clue RT for Rotten Tomatoes, a website that reviews movies that, as the name implies, do not win gold medals. This has nothing to do with marathons (other than maybe a movie watching marathon) but OP said in the comments that a lot of the words are "gibberish" to "maintain the flow". The next line seems to confirm that this is in fact about movies.

But the sums of my life is the sums of all the races I ran and all the movies I saw.

This confirms that the previous clue was about movies and confirms that the answer is AL + BE + RT = ALBERT

And that gives me the energy to go ahead and fulfil my dreams.

Albert Einstein discovered the famous equation $E = mc^{2}$ which relates energy to mass and the speed of light.

Who am I? I've been asking the same to myself for over a hundred years lmao

Albert Einstein was born in 1879 and has had over a century to question his identity. If this answer is correct then I hope it helps mend his heart and reaffirm his identity.

The hints:

Albert Einstein had White hair and did physics.