[1] Modal particles have various functions, including adding emotion or emphasis,[2] or to express how sentence content is grounded in common knowledge between the speaker and participants.
The German particle ja is used to indicate that a sentence contains information that is obvious or already known to both the speaker and the hearer.
The sentence Der neue Teppich ist rot means "The new carpet is red".
Der neue Teppich ist ja rot may thus mean "As we are both aware, the new carpet is red", which would typically be followed by some conclusion from this fact.
In Dutch, modal particles are frequently used to add mood to a sentence, especially in spoken language.