Kaivalya mukti is a state of solitude, loneliness, isolation and detachment. Word is Kaivalya is its roots in the Sanskrit word "kevala ,केवल ", meaning alone isolated. It is a separation of Self or Soul of the yogi from prakriti or nature or out side world . This is final stage of enlightenment that a yogi can reach, also in other words called as moksha.
So Kaivalya mukti is a detachment and independence from our relationships, egoism, attraction, desires and birth and death cycle . A yogi must perform austerities or Tapas , yoga and and achieve self control . The yogis who reach this state is called a Kevalin .
The Yoga Sutras of Patanjali as well as various Muktika Upanishads do talks about a yogi who achieved kaivalya and is independent from all bonds and this samsara
of shukla Yajurveda - Chapter XIII (thirteenth khaṇḍa) is giving us the description of Kaivalya state and a Yogi in that state.
बाल्येन तिष्ठासेद्बालस्वभावोऽसङ्गो निरवद्यो मौनेन पाण्डित्येन
कैवल्यमुक्तं निगमनं प्रजापतिरुवाच महत्पदं
ज्ञात्वा वृक्षमूले वसेत कुचेलोऽसहाय एकाकी |
आप्तकामो निष्कामो जीर्णकामो हस्तिनि सिंहे दंशे मशके नकुले |
सर्पराक्षसगन्धर्वे मृत्यो रूपाणि विदित्वा न बिभेति कुतश्चनेति वृक्षमिव
न कुप्येत न कम्पेतोत्पलमिव
तिष्ठासेच्छिद्यमानोऽपि न कुप्येत न कम्पेताकाशमिव
न कुप्येत न कम्पेत सत्येन
The wise man should conduct himself like a lad, with the nature of a
child, without company, blameless, silent and wise and without
exercising any authority. This description of Kaivalya is stated by
Prajāpati. Having found with certitude the supreme seat, one should
dwell under a tree with torn cloths, unaccompanied, single and engaged
in samādhi. He should be longing after the attaining of Ātmā and
having attained this object, he is desireless, his desires have
decayed. He fears none, though he finds the cause of death in such as
elephants, lions, gadflies, musquitoes, ichneuma, serpents, Yakṣas,
Rākṣasas, and Gandharvas. He will stand like a tree. Though cut down,
he will neither get angry nor tremble. He will stand (or remain) like
a lotus. Though pierced, he will neither get angry nor tremble. He
will stand like ākāś; though struck, he will neither get angry nor
tremble. He will stand by Satya (truth), since Ātmā is Satya.
In his Yoga sutras -Part 4 - kaivalya Pada - Patanjali tells us about what happens when a yogi detaches himself from prakriti or this samsara. In this state a yogi isolates himself from prakriti and her gunas , then they don't have any purpose to serve for the Atman or self , which a yogi realised , they merge with nature. And self alone remains , and shines forth as pure consciousness , ultimate truth as its true nature.
पुरुषार्थशून्यानां गुणानां प्रतिप्रसवः कैवल्यं स्वरूपप्रतिष्ठा वा
चितिशक्तिरिति ॥ 34||
Puruṣārthaśūnyānāṃ guṇānāṃ pratiprasavaḥ kaivalyaṃ svarūpapratiṣṭhā vā
Since the gunas no longer have any purpose to serve for the Atman,
they resolve themselves into Prakriti. This is liberation. The Atman
shines forth hilts own pristine nature, as pure consciousness.
And this is second explanation of a Yogi in his Kaivalya state of above sutra by Patanjali.
The enlightened Yogi: Such an enlightened Yogi is purely spontaneous,
with no actions whatsoever being motivated by the inner drives of
samskaras and karma. One hundred percent of actions are from the
here-and-now response to the needs of the moment, in relation to the
service of other beings. This is easy for such a yogi, as there is no
I and no other; it is all a constant flow of pure, undivided
consciousness (purusha), that only seems to play, here, there, and
In Kaivalya mukti no new karmas are generated ,since he is completely detached from the world , his sanskaras or memories which are responsible for birth and death cycle vanishes. a yogi completely gets free from differences i.e. dvaita and he experience complete unity with Brahman . And dwells in that state doing karma's just for others , having no desires of fruits.