The title is a play on words between his nickname "Mr. Braun", and the scientific phenomenon known as Brownian motion.
The story is set in 1996-2000, and concentrates on the relationship between a young Yuugo Tennouji and Suzu Hashida, 22 years after she leapt back to 1975 to secure the IBN 5100. Yuugo, who was then working actively as a high ranking Rounder, is tasked with a mission (given by the commitee of 300) to travel to Japan, Akihabara, and investigate Suzu and also locate the IBN 5100. During his mission, Yuugo develops strong relationships with both Suzu and Tsuzuri Imamiya (whom he would later marry and give birth to Nae Tennouji with). This was seen by Yuugo's higher-ups as a problem, thus he was faced with the burden of constant coercion from the Rounders, and safety of his newly formed family and friends.
The story begins on the 4th July, 2010, following the events of a regular day; whereby Okabe and the Labmems experiment with their new gadget, and Tennouji threatens to raise the rent. After the ordinary series of events, Nae comes to visit her father, and Yuugo decides to take her on a trip to visit Suzu's gravestone. He remarks that "Nae won't remember that place now, I think".
We look back in time to the 22nd December, 1996 and see a younger Yuugo Tennouji. His new mission is to go to Japan and investigate a strange woman, Hashida Suzu. Her past prior to 1975 is blank, arousing suspicion from SERN and the Rounders. She is the suspected hacker of SERN, and may also possess one of the few remaining IBN 5100s. Yuugo meets her by chance just after his arrival in Akihabara, as she mistakenly knocks him out for thinking he was harassing her student, Tsuzuri Imamiya.
After the confusion is cleared, Suzu - who was then working as a university professor and landlord (of the building where the FGL would be located) - offers a cheap rent room as an apology to Yuugo. After realising who she had just met, Suzu, imitating Okabe, remarks to herself that this must be "the choice of the Steins;Gate", and continues to help Yuugo.
Yuugo initially takes advantage of his quick relation to Suzu as a means to accelerate his mission, however he begins to form genuine bonds to both Suzu and Tsuzuri, complicating his job as a Rounder.